Continuity and Differentiability Class 12

Ex5.1,1 teachoo.com
Prove that the function f (x) = 5x - 3is continuous atx = 0, at
x =-B3andatx =5
Given f(x) = 5x-3
At x=0
f(x) is continuous at x = 0 if

lim f(x) = f(0)

Ex 5.1, 2 teachoo.com
Examine the continuity of the function f (x) = 2x? -— 1 at x =3.
f (X) is continuous at x = 3 if
lim f@) = f(@)
L.H.S R.H.S
lim f(x) f3)
= 2
= lim (2x?-1) (3-1
” =2x9-1
Putting x = 3
= 18-1
=2(3-1
= 17
=2x9-1
= 17
Since, L.H.S = R.H.S
Hence, f is continuous at x = 3

teachoo.com

Ex 5.1, 3
Examine the following functions for continuity.
(a) fx)=x-5
fix)=x-5
To check continuity of f (x),
We check it’s if it is continuous at any point x = ¢
Let c be any real number
fis continuous atx =c

if lim f@X) = f(©)

x-c

Examine the following functions for continuity. (b) f (x) = 1/(𝑥 − 5) , x ≠ 5 f (x) = 1/(𝑥 − 5)
At x = 5
f (x) = 1/(5 − 5) = 1/0 = ∞

Examine the following functions for continuity. (c) f (x) = (𝑥^(2 )− 25 )/(𝑥 + 5), x ≠ –5 f (x) = (𝑥^(2 )− 25 )/(𝑥 + 5)
Putting x = –5
f (−5) = (〖(−5)〗^(2 )− 25 )/(−5 + 5) = (25− 25 )/(−5 + 5)
= 0/0 = Undefined

Examine the following functions for continuity. (d) f (x) = |x – 5| f(x) = |𝑥−5|
= {█((𝑥−5), 𝑥−5≥0@−(𝑥−5), 𝑥−5<0)┤
= {█((𝑥−5), 𝑥≥5@−(𝑥−5), 𝑥<5)┤

Ex 5.1, 4 teachoo.com
Prove that the function f (x) = x” is continuous at x = n, where n
is a positive integer.
f (x) is continuous at x = n if
lim G2) = f@)
LHS RHS
lim f@) f(n)
= lim x” =n"
xon
Putting x =n
= n™
Since, L.H.S = R.H.S
-. Function is continuous at x =n

Ex 5.1, 5 teachoo.com
. . x,ifx<s1
Is the function f defined by f(x) = {e ifx>1
continuousatx = 0 ?Atx = 1 PAtx = 2?
. x,ifx<1
Given f(x) = { ifx>1
Atx=0
Forx=0,
f(x) =x
Since this a polynomial
It is continuous
-. f(x) is continuous for x = 0

Ex 5.1, 6 teachoo.com
Find all points of discontinuity of f, where f is defined by
2x+3, ifx<s2
fay = [re ifx>2
_ (2x43, ifx<s2
FG) = eae ifx>2
Since we need to find continuity at of the function
We check continuity for different values of x
« Whenx=2
« Whenx<2
« Whenx>2

Ex5.1,7 teachoo.com
Find all points of discontinuity of f, where f is defined by
|x] +3, ifx<-3
f@= —2x, if-3<x<3
6x + 2, ifx>3
Since we need to find continuity at of the function
We check continuity for different values of x
* Whenx<-3
* Whenx=-3
« When-3<x<3
* Whenx=3
* Whenx>3

Ex 5.1, 8 teachoo.com
Find all points of discontinuity of f, where f is defined by
Ix].
—, ifx#0
foy=}e' f
0,ifx=0
Since we need to find continuity at of the function
We check continuity for different values of x
« Whenx=0
« Whenx>0
« Whenx<0
Case 1: When x=0 Since there are two different
functions on the left & right of 0,
fod is continuous at x = 0
we take LHL & RHL.
if LH.L= R.H.L= £(0)

Ex 5.1, 9 teachoo.com
Find all points of discontinuity of f, where f is defined by
x
—,ifx<0
f(x) = |x] f
-1,ifx20
Since we need to find continuity at of the function
We check continuity for different values of x
* Whenx=0
* Whenx>0
* Whenx<0
Since there are two different
Case 1: When x=0
functions on the left & right of 0,
f(x) is continuous at x = 0
we take LHL & RHL.
if LH.L= R.H.L= f(0)

teachoo.com
Ex 5.1, 10
Find all points of discontinuity of f, where fis defined by
x+1, ifx2>1

rod={ e+1,ifx<1
Since we need to find continuity at of the function
We check continuity for different values of x

* Whenx=1

* Whenx<1

* Whenx>1

teachoo.com
Ex5.1, 11
Find all points of discontinuity of f, where f is defined by
8-3, ifx<2

ro) ={ x41, ifx>2
Since we need to find continuity at of the function
We check continuity for different values of x

¢ Whenx=2

¢ Whenx<2

« Whenx>2

teachoo.com
Ex 5.1, 12
Find all points of discontinuity of f, where f is defined by
x-41, ifx<s1
roy= {7 ifx>1
Since we need to find continuity at of the function
We check continuity for different values of x
* Whenx=1
* Whenx<1
* Whenx>l

teachoo.com
Ex 5.1, 13
Is the function defined by
fx <1

reo={ ee PS
a continuous function?
Since we need to find continuity at of the function
We check continuity for different values of x

° Whenx=1

« Whenx<1

* Whenx>d

Ex5.1, 14 teachoo.com
Discuss the continuity of the function f, where f is defined by
3, ifO<x<1
f=, 4 ifl<x<3
5, if 3<x< 10

Since we need to find continuity at of the function
We check continuity for different values of x

* WhenOsx<1

* Whenx=1

* When1<x<3

* Whenx=3

* When3<xs10

teachoo.co
Ex 5.1, 15 "
Discuss the continuity of the function f, where f is defined by
2x, ifx <0
f= 0,if0O<x<1
4x, ifx>1
Since we need to find continuity at of the function
We check continuity for different values of x
« Whenx<0O
« Whenx=0
* WhenO<x<1
«© Whenx=1
«© Whenx>l

Ex 5.1, 16 teachoo.com
Discuss the continuity of the function f, where f is defined by
F; ifx<-1
f(x) =< 2x, if -1<x<1
2,ifx>1

Since we need to find continuity at of the function
We check continuity for different values of x

¢ Whenx<-1

¢ Whenx=-1

¢ When-1<x<1

¢ Whenx=1

¢ Whenx>1

Ex 5.1, 17 teachoo.com
Find the relationship between a and b so that the function f
defined by

_faxt+1ifxs3
FG) = (rtsipess
is continuous at x = 3.
Given function is continuous at x = 3

Since there are two different

f(x) is continuous at x = 3 functions on the left & right of 3,
if LH.L= R.H.L= f(3) we take LHL & RHL.
if Jim f(x) = lim, f@) = £@)

Ex 5.1, 18 teachoo.com
For what value of A is the function defined by

_ (AG? — 2x), ifx <0
ro = ifx>0
continuous at x = 0? What about continuity at x = 1?
Atx=0

Since there are two different
f(x) is continuous at x = 0 functions on the left & right of 3,
if LH.L= R.H.L= (0) we take LHL & RHL.
if if lim f@= lim f@)=f(0)
x30 x30

teachoo.com
Ex 5.1, 19 (Introduction)
Show that the function defined by g (x} = x — [x] is discontinuous
at all integral points. Here [x] denotes the greatest integer less
than or equal to x.
Greatest Integer Function [x]
Going by same Concept
Value of
Value of x | Greatest Integer Value of
bd Value of x | Greatest Integer
3 3 Ix]
c c
3.1 3
ct=cth c
3.9999 3
c=c-h e-1
4 4
4.0001 4

Ex 5.1, 20 teachoo.com
Is the function defined by f (x) = x? —sin x + 5 continuous at x = 1?
f(x)=x? -sinx45

Let p(x) = x*, q(x) =sinx & r(x) =5

p(x) = x? is continuous as it is a polynomial

q{x) = sin x is continuous at all real numbers

r(x) =5 is continuous as it is a constant function

By Algebra of continuous functions,

If px) , g(x) & r(x) all are continuous at all real numbers

then f(x) = p(x) — q{x) + r(x) is continuousat all real numbers
f(x) = x? —sin x +5 is continuous at all real numbers.

Thus, f (x) is continuous at x = 7

Ex 5.1, 21 teachoo.com
Discuss the continuity of the following functions:

(a) f («) = sinx + cosx

f (x) = sinx +cosx

Let p(x) = sinx & q(x) = cosx

We know that sin x & cos x both continuous function

- p(x) & q(x) is continuous at all real number

By Algebra of continuous function

If p(x) & q(x) are continuous for all real numbers

then f(x) = p(x) + q(x) is continuous for all real numbers
« f(x) = sinx + cosx continuous for all real numbers

teachoo.com

Ex 5.1, 22 (i)
Discuss the continuity of the cosine, cosecant, secant and
cotangent functions.
Let f(x) = cosx
To check continuity of f (x),
We check it’s if it is continuous at any point x = c
Let c be any real number
fis continuous atx = c if

if LH.L = RH.L= f(c)

ie. lim f(x) = lim f(x) = fc)

Koc xXoC

Ex 5.1, 22 (ii) teachoo.com
Discuss the continuity of the cosine, cosecant, secant and
cotangent functions.
Let f(x) = cosec (x)
1
FO) = ae
Let p(x) = 1&q(a) =sinx
Since, p(x) is a constant,
* p(x) is continuous.
We know that sin x is continuous for all real numbers
+. q(x) is continuous.

Ex 5.1, 22 (iii) teachoo.com
Discuss the continuity of the cosine, cosecant, secant and
cotangent functions.
Let f(x) = secx
1
f() = wosz
Let p(x) = 1& q(x) =cosx
Since, p(x) is a constant,
«. p(x) is continuous
We know that cos x is continuous for all real number
«. q(x) is continuous

Ex 5.1, 22 (iv) teachoo.com
Discuss the continuity of the cosine, cosecant, secant and
cotangent functions.
Let f(x) = cot x

CcOSX
f@) ~ sin x
f (x) is defined for all real number except where sin x =0
ie. x=n
Let p(x) = cosx & q(x) =sinx
We know that sin x & cos x is continuous for all real number
-. p(x) & q (x) are continuous functions

Ex 5.1, 23 teachoo.com
Find all points of discontinuity of f, where
sin x
—, ifx<0
foal iw
x+1, ifx20
Since we need to find continuity at of the function
We check continuity for different values of x
* Whenx<0
« Whenx=0
* Whenx>0

Ex 5.1, 24 teachoo.com
Determine if f defined by
1
x? sin-, ifx#0
feoga{esne Ff
0, ifx=0
is a continuous function?
Since we need to find continuity at of the function
We check continuity for different values of x
« Whenx#0
« Whenx=0

Ex 5.1, 25 teachoo.com
Examine the continuity of f, where fis defined by
sinx —cosx, ifx+#0
roo = {Re ifx=0
_ fsinx —cos x, ifx#0
ro = {2 if x=0
Since we need to find continuity at of the function
We check continuity for different values of x
« Whenx#0
« Whenx=0

Ex 5.1, 26 teachoo.com
Find the values of k so that the function f is continuous at the
indicated point
KR cos x . cia
—, ifx#s
fe) =\7—** 2 atx=t
3, ifx=n 2
2
Given that function is continuous at x = 5
f is continuous at = 5
if LH.L=RHL= f (5)
. . . we
ie. lim f(x) = lim, f= f(§)
x>*D xD

teachoo.com
Ex 5.1, 27
Find the values of k so that the function fis continuous at the
indicated point
kx?, ifx<2

ro = {fF ifx>2 at x=2
Given that function is continuous at x = 2
f is continuous atx =2

if LH.L=R.H.L= f(2)

ie. lim f@) = Jim f@) = f@)

Ex 5.1, 28 teachoo.com
Find the values of k so that the function f is continuous at the
indicated point
kx+1, ifxen

roo = {et ifx>1 at x=
Given that function is continuous atx = 7
f is continuous atx = 7

If L-H.L = R.H.L = f (77)

ie. tim f(x) = Jim, f@) = fo)

teachoo.com

Ex 5.1, 29
Find the values of k so that the function fis continuous at the
indicated point

_fkxt+1, ifxss _
ro = {grt ifx>5 at x=5
Given that function is continuous atx = 5
f is continuousatx =5

if L.H.L=R.H.L = f(5)

ie. lim f(x) = lim f(x) = f(S)

xo5” xo5t

Ex 5.1, 30 teachoo.com
Find the values of a and b such that the function defined by
5, ifx<2
f@)=sfax+b, if2<x<10
21, ifx>10
is a continuous function
Since f(x) is a continuous function,
It will be continuous for all values of x
Atx=2
A function is continuous at x = 2
if LH.L= R.H.L = f(2)
ie. jim fG) = lim f)= FQ)

Ex 5.1, 31 teachoo.com
Show that the function defined by f(x) = cos(x*) is a continuous
function.
f (x) = cos(x’)
Let g(x) = cosx & h(x) =x?
Now,
goh(x)
= g(h(x))
= g(x?)
= cos(x)
= f(x)
Hence, f(x) = goh(x)

Ex 5.1, 32 teachoo.com
Show that the function defined by f (x) = |cosx| isa
continuous function.
f(x) =|cosx|
Let g(x) = |x| & h(x) =cosx
Now,
goh(x)

= g(r(x))

= g(cosx)

= [cos x|

= f(x)
Hence, f(x) = goh(x)

Ex 5.1, 33 teachoo.com
Examine that sin | x | is a continuous function.
f@) = sin |x|
Let g(x) = sinx & h(x) = |x|
Now,
goh (x)
= g(A@))
= g(lxl)
= sin |x|
= f(x)
Hence, f(x) = goh (x)

Ex 5.1, 34 teachoo.com
Find all the points of discontinuity of f defined by
FO) = lxl- [x +11.
Given f(x) = [x|- |x +1I.
Here, we have 2 critical points
x=Oandx+1=0
ie. x=0,andx=-1
So, our intervals will be
* When x<—-1
¢ When -1<x<0
¢ When x>0

Ex 5.2, 1 teachoo.com
Differentiate the functions with respect to x sin(x? + 5)
y =sin (x24 5)
We need to find derivative of y, w.r.t.x
dy _d(sin@’ + 5))
dx dx
d(x? +5)
= 2 oe
cos (x? + 5) x -
d(x’) a)
= 2
cos (x? + 5) x( ix + x
= cos (x2 + 5) x (2x2-1 +0)
=cos (x? + 5) x 2x
=2xcos (x? + 5)

teachoo.co
Ex5.2, 2 ma
Differentiate the functions with respect to x
cos (sinx)
Let y= cos (sin x)
We need to find derivative of y, w.r.t.x
_ dy _ d(cos (sin x))
eae = dx
_ . . d(sin x)
=—sin(sinx). x
=— sin(sin x) . cos x
=— cosx sin (sin x)

EX 5.2, 3 teachoo.com
Differentiate the functions with respect tox sin(ax + b)
Let y=sin (ax + b)
We need to find derivative of y, w.r.t.x
dy _ d(sin(ax + b))
‘dx dx
_ d(ax+b)
=cos {ax + b) x an
_ d(ax) a“)
=cos (ax + b)x (= + val
=cos {ax + b).(a+0)
=acos(ax + b)

Ex 5.2, 4 teachoo.com
Differentiate the functions with respect to x
sec (tan ( yx ))
Let y=sec(tan /x)
We need to find Derivative of y
ie. y’ = (sec (tan /x))’
= sec (tan /x) tan (tan x) (tan yx )!
= sec (tan Vx) tan (tan x). (sec? Vx. (/x)’)
= sec (tan/x) tan (tan/x). sec? x x a
_ Sec(tan yx) tan(tan /x) sec? Vx
Te Oo

Ex5.2,5 teachoo.com
i +B
Differentiate the functions with respect to x : sin (ax +)
cos (cx +d)
sin (ax + b)
let y= cos (cx +d)
Letu=sin (ax+b) & v =cos (cx +d)
wyott
oo y = v
We need to find derivative of y w.r.t.x
dy (“)
dx \v
dy _ulv—v'u
dx pe

Ex 5.2, 6 teachoo.com
Differentiate the functions with respect to x
cos x° . sin? (x5)
Let y= cos x3. sin? (x°)
Letu=cosx? & v=sin? (x°)

“y=uv
We need to find derivative of y w.r.t.x

y’ = (uv)’

su'vt+v'u

Ex 5.2,7 teachoo.com
Differentiate the functions with respect to x
2,/ cot (x?)
Let y= 2,/cot (x?)
We need to find derivative of y w.r.t.x
y' = (2/cot (x?) )
1 '
=2 x ———— . . (cot (x?
x 2 cot (x2) (co (x )
= —_. (-cosec? (x?)) . (x?)
vy cot (x?)
= —. (—cosec? (x*)) .2x
vy cot (x?)

Ex 5.2,8 teachoo.com
Differentiate the functions with respect to x
cos (x)
Let y= cos (Vx)
We need to find derivative of y w.r.t.x
_, dy _ a(cos yx)
eo = ae
_ ‘ d(yx)
= -sinJx . dx
= -siny¥. 2s
= —sinyx . 5 Vz
_ sin yx
“TE

Ex 5.2,9 teachoo.com
Prove that the function f given by
f= |x-1,x*x ER
is not differentiable at x = 1.
f(x) = |x-1|
_f@-1, x%x-120
~ |-(«-1), x-1<0
_f@-1, x21
“j-(«-1), x<1
Now,
f(x) is a differentiable at x = 1 if
LHD = RHD

Ex 5.2, 10 (Introduction) feachoo.com
Prove that the greatest integer function defined by
fi = [x],0<x<3
is not differentiable atx = 1 andx= 2.
Greatest Integer Function [x] Going by same Concept
id So
Value of x Value of x
Integer [x] Integer [x]
1 1 1 1
1.0001 1 1th 1
0.9999 0 1-h i)
2 2 2 2
2.0001 2 2th 2
1.9999 1 2-h 1

Ex 5.3, 1 teachoo
. dy. _ oa:
Find a i 2x + 3y = sinx
Given
2x + 3y = sinx
Differentiating both sides w.r. t. x
d(2x+3y) _ d(sinx)
dx ~ dx
d(2x) + dy) _ d(sin x)
dx dx ~ dx
2 4 + 320) = d(sin x)
dx dx dx
2+3@ =cosx
dx
3 Ys cosx-2
dx
wet (cos x - 2)
ax 3

Ex 5.3, 2 teachoo.com
. dy. _ a
Find in, 2x + 3y = siny.
2x + 3y = siny
Differentiating both sides w.r. t.x
d@2x+3y) _ d(siny)
dx 7 dx
ax) + aGy) = a (sin y) (Derivative of sin x is cos x)
dx dx dx
pyaad + 3dQy) _ d (sin y) 2
dx dx dy dx
d d
2+3— =cos yx oy
dx dx
d d
cosy X oy 39 _ 9
dx dx

teachoo.com
Ex 5.3, 3
dy,
Find in, ax + by? = cosy
ax + by? = cosy
Differentiating both sides w.r.t.x
d(ax +by’) _ d(cosy)
dx ~ dx
d(ax) + d(by”) _ d(cosy)
dx dx dx
dx a4) _ a
a +b Gz > an OSY
ath 122 4 = A(oosy) 5 ay
dx dy dx dy
ath 122 x = A(oosy) , ay
dy dx dy dx
dy _o dy
a+b.2yx a SNY

Ex 5.3, 4 teachoo.com
Find “ in, xy + y? = tanx + y
xy + y? = tanx+y
Differentiating both sides w.r.t.x
d(xy+y*) _ d(tanx+y)
dx ~ dx
d(xy) + dQ”) _ d (tanx) 47@ oy)
dx dx dx dx
Using product rule in (xy)! =x'y+y'x
d(x) d(y) ) d(y?) _d(tanx) | dy
—, x) tS tS
( dx y + dx x dx dx dx
dy _dQ”) | dy 24,
yews xa +
1 yt x dx dx dy sec’ x dx

Ex 5.3,5 teachoo.com
dy.
Find —in, x? + xy + y* = 100
dx
x? + xy + y* = 100
Differentiating both sides w.r.t.x .
d(x? +xy+y*) _ d (100)
dx ~ dx
d(x’) 4 d(xy) 4 diy’) _ a (100)
dx dx dx dx
As (x")! = nx"
& derivative of a constant is zero
-1, dy) , dQ”) | ay
2-1 “Ye
2x + ax + dx x dy 0
d@y) , dQ”) | dy_
2x + ax + dy XO

teachoo.co
EX 5.3, 6 CACHOO.LOM
ng We 3 2 2 3-
Find =~ in, x + x*y + xy? + y? = 81
x4 x’y + xy? + y3 = 81
Differentiating both sides w.r.t.x .
d(xitx*ytay?+y*) _ d(81)
dx ~ dx
ace) | d@’y) | dy’) | 46°) _ oe
“a tae tae t+ ae = 0 (Derivative of constant 0)
3-1, a@’y) | dy) | dO), dy _
3x + dx + dx + dx * ay 9
2, ay), day’) | do), ay_
3x" + dx + dx + dx * dx =0
2, 4@’y) . atxy’) 3-1
3x + ra + ie +3y ant 0
2, ay) , dtxy’) 2 dy _
3x* + a + ke +3y ax” 0

Ex 5.3, 7 teachoo.com
dy
Find in. sin? _
ind a it sin’ y + cos xy =1
sin? y + cos xy =
Differentiating both sides w.r.t.x .
d(sin’y+cos xy) _ d(x)
dx ~~ dx
d (sin? d(cos x
asin’ y) + a (cos xy) =0 (Derivative of constant is zero)
dx dx
Calculating Derivative of sin? y & cos (xy) separately

teachoo.com
Ex 5.3, 8
ind i in2 2y =
Find ax in, siné¢x + cos*y = 1
sin?x + cos?y = 1
Differentiating both sides w.r.t.x .
d(sin’x +cos*y) _ d(1)
dx dx
an? 2
asin’ x) , d(cos"¥) _ 9 (Derivative of constantis 0)
dx dx
Calculating Derivative of sin? x & cos* y sepretaly
Finding Derivative of sin? x
d (sin? x) =2sin2-1 x. d(sin? x)
dx dx
=2sinx. atin x)
dx

Ex 5.3,9 teachoo.com
. dy. cin 1 2x
Find (in, y= sin (Z 2x? )
rs 2x
y=sin (SS)
Putting x = tan 8
soe 2x
_ ont { 2 tan@
y=sin (; + a)
. . 2tan@
y=sin-? (sin26) (since sin 20 = ae)
y = 20
Putting value of 8 =tan-1x Since x= tan ®
y = 2tan tx « tan-'x=6

Ex 5.3, 10 teachoo.com
ind xi = 3 (==) ae 1
Find an y =tan tae}? a" <e
=tant (==)
y= 1- 3x"
Putting x = tan 8
- tan-1 C tan @ — tan ~)
y= tan 1-3tan’@
3 tan @-tan *@
y=tan™* (tan3 6) (tan 30 = aca)
y = 30
Putting value of 8 =tan™! x
- tan-'x=0
y = 3(tan™'x) an_*

teachoo.com
Ex 5.3, 11
Find in, y = cos (=) O<x<1
ind in, y=cos*(—— a), x
_ 1 (1-%?
Putting x = tan 8
_ -1 ( 1-tan 76 )
y= cos 1+ tan7@
1-tan’¢@
y = cos"! (cos 26) (cos 20 = 1+ tan? a)
y =20
Since x = tan 8
Putting value of 8 =tan7* x » tan-?x=0
y =2 (tan x)

Ex 5.3, 12 teachoo.com
. ay, _, (1-x?
Find ——in, y= sin 1 (=), O<x<1
= sin? (=)
y 1+ x?
Putting x = tan 8
= sin (2 — tan? )
y= 1+ tan? @
We know that
y=sin~* (cos 20) 1 tan 26
cos 20 =——_
14+ tan°é
y =sin"1+ (sin G - 20))
we
y = 77 26

Ex 5.3, 13 teachoo.com
a 2
Find — in, y=cos* (=) ,-i<x<1
dx 1+ x’
_ -1 2x
y=cos* (>)
Letx = tan@
= cost ( 2 tan @ )
ye 1+ tan’@
y = cos (sin 28)
y =cost (cos G - 26))
y => - 20
; 1 Since x = tan 8
Putting value of 8 =tan*x ~ tan>x=8
ye 5 -2tan"'x

teachoo.com
Ex 5.3, 14
Find 2 in, y=sin (2x Vv1-x?), ~BEKKG
y=sin? (2x v1—x?)
Putting x = sin@
y=sin? (2 sin@ v1 —sin26)
y=sin (2 sin 8 Vcos26) (1 — sin?@ = cos?@)
y =sin (Qsin® cos 6)
y =sin" (sin2 @) (sin 28 =2cos@ sin@)
y = 20
Putting value of 8 = sin x
« sin->x=0

teachoo.com
Ex 5.3, 15
ind i = sect (=) _
Find ax in, y=sec mea)! O< X<75
1
= cect
_ 1
secy= a-4
1t. 1
cosy ~ 2x2 =1
cosy =2x*-1
y=cos 1(2x? — 1}
Putting x = cos@
y =cos'(2cos?@ — 1)

teachoo
Ex 5.4, 1
. . . e*
Differentiate w.r.t.x in, —
sinx
eX
lety ~ sing
Letu =e*& v = sinx
. u
Differentiating both sides w.r. t. x
U
ay) _ a3)
dx dx
dy (uv -v'w
dx v2

teachoo.com
Ex 5.4, 2 (Method 1}
Differentiate w.r.t.xin, es **
int
lety = es *
Differentiating both sides w.r. t.x
a(y) a(esin™ *)
dx dx
oo a(e*)
dy _ gsin*x a(sin~* x) (© = e*)
dx dx
d 1 a(sin7! x) 1
ay = esin"*x (=) dx = V1—x2
ax ° "AW1 — x2 .
d(y) esinet x
dx V¥1—-x2

Ex5.4.3 teachoo.com
x 5.4,
Differentiate w.r.t.x in, e*
Lety = ex
Differentiating both sides w. r. t.x
3
ay) _ a(e*)
ax ax
a(e*)
a 3 The =
ed = ex . a(x*) ( ax e )
ax ax
d ‘th
PY _ gx? 342 (4s, “2 = nx" +)
dx dx
dy 3
— = 3x" e*
dx

Ex 5.4, 4 teachoo.com
Differentiate w.r.t.x in, sin (tan7+e-*)
Let y =sin (tan-te~*)
Differentiating both sides w.r. t. x
y’ = (sin(tan-1 e~*))’
= cos (tan-t e-*) x (tan7t e~*)’ ((sin x)’ = cos x)
-1e* t -xy ((tan-* xy= =)
= cos (tan7* e ) xyes “Ce ) =e
= cos (tan-1 e~*) x ——~__ x -e-*
1+ (e-*)?

Ex 5.4,5 teachoo.com
Differentiate w.rt. xin, log (cos e~)
Let y = log (cos e*)
Differentiating both sides w.r. t. x
d(y) _ d(log (cos e*))
dx dx
x d 1
dy __1 d(cos e*) (4s a (log x) = *)
dx cose*’ dx
1 d(e*) ( da ; )
= —— (~sine*).—— As — (cosx) = —sinx
cosex' 6 sine). in wf )
1 . ate*) )
- — (— x x Be) = 9X
cosex' 6 sine*).e ‘Cn e
— sin e~
=_- —__ x
~ cose® * e
= —tane* .e*
= —e*. tane*

Ex 5.4, 6 teachoo.com
Differentiate w.r.t.x in, e7+ er +e ter + eX
Let y= e*+ ee +te®
yrers oF te% er + eX
Differentiating both sides w. 7. t.x
Z 3 4 Ss
a(y) a(eX + eX + eX tex" + e*°)
dx dx
2 3 A 5
dex d(e* d(e* d(e* d(e*
_ acer) ae") ale") ae!) ale”)
dx dx dx dx dx
2 3 4 5
= et 4 er? 4") ter 4) eer? 4) ter 4)
dx dx dx dx
2 3 A 5
= e* +2xe* + 3x7 e* +4x3 e* + 5x4 e*

Ex 5.4, 7 teachoo.com
Differentiate w.rt. xin, v eve, x>0d
Lety =V¥V ev*
Differentiating both sides w. r. t.x
f
v= (Se)
1 i
re x eve
y'= ae x (e%*)
1
y 2Vvev™
1 1
roth oy eV yx
y 2Vvev™ 2vx

teachoo.com
Ex 5.4, 8
Differentiate w.rt. xin, log(logx),x>1
Let y = log (log x)
Differentiating both sides w. r. t.x
d(y) _ d(log (log x))
dx dx
dy __1 , dog x) (4s £ (og x) = +)
dx log x dx dx *
dy 1 1 d 1
Ss = Ke As — (log x) = =
dx logx *% ( ax (log x) )
dy _ 1
dx xlog x

Ex 5.4, 9 teachoo.com
Differentiate w.r.t. x in, “ x>2
log x
lety = cosx
y ~ logx
Let u = cosx & v =logx
_ ou
ee y = 7
Differentiating both sides w. r. t. x
/
1 *)
y(¢
dy _ ulv —v!u
dx v2
dy _ (cos x)' logx — (log x)'.cosx
dx (log x)

Ex5.4, 10 teachoo.com
Differentiate w.rt.x in, cos(logx + e*), x > 0
Let y =cos(logx + e*)
Differentiating both sides w.r. t. x

y’ =(cos(logx + e*))’

y’= —sin (logx + e*) Cogx + e*)’

y’= —sin (logx + e*) (Cogx)’ + (e*)’)

ro_ig xy fi x
y’ =— sin (logx + e”) C te )
yor G + e*) sin (logx + e*)

Ex 5.5, 1 teackoo
Differentiate the functions in, cos x .cos 2x .cos 3x
Let
y= cosx.cos 2x .cos 3x
Taking log both sides
log y = log (cos x. cos 2x . cos 3x)
log y = log (cos x) + log (cos 2x) + log (cos 3x)
Differentiating both sides w.7r. t. x.
dllogy) d(log(cos x) + log (cos 2x) + log (cos 3x))
ax 7 dx
d(log y) (2) _ d(log (cos x)) + d(log (cos 2x)) + d(log (cos 3x)
dx dy. ~ dx dx dx

Ex 5.5, 2 teachoo.com
. . - («-1)@ - 2)
Differentiate the functions in, @-De-D@—5)
= @-1)@-2)
let y= Nee -N@—5)
i
_ (x - 1) - 2) 2
y= (q ~2)a-)a- 5)
Taking log both sides
i
_ @-1)@- 2) 2
logy = log (q “DE -De 5)
1 @-1)@—-2) b
== |} As l = bl
logy 5 log (Sie oes) (As log(a”) b log a)

Ex 5.5, 3 teachoo.com
Differentiate the functions in, (log x)°°S*
Let y = (logx)°°S*
Taking log both sides
logy =log (logx)*°**
logy = cos x .log (log x) (As log(a’) = bloga)
Differentiating both sides w.7r. t. x.
d(logy) _ d(cosx.log (log x))
dx ~ dx
d(log y) (2) _ a(cosx . log (log x))
dx dy. ~ dx
d(log y) (2) _ d(cos x. log (log x))
dx dx} dx

teachoo.com

Ex 5.5, 4
Differentiate the functionsin, x* - 25™*
Let y=x* - 28inx
Let u=x* ,v = 25In*

yru-v
Differentiating both sides w. r. t. x.

dy d(u-v)

dx dx

dy _ du dv

dx dx dx

teachoo.com

Ex5.5,5
Differentiate the functions in, (x + 3)*.(x + 4)?.(x« + 5)4
Let y=(x + 3)*.(x + 4)°.(x + 5)4
Taking log both sides
logy= log (( + 3)?.( + 4)3.(@ + 5)4)
logy = log (x + 3)° +log (x + 4)? +log (x + 5)*
logy= 2log (x + 3) +3log(x + 4)+4log (x + 5)
Differentiating both sides w.r.t. x.
dQogy) _ d(2log(v+3) + 3log (x +4) + 4 log (x +5))

dx ~ dx

Ex 5.5, 6 teachoo.com
1\* i
Differentiate the functions in, (x + =) + x(t + 2)
x 1
Let y= (x+=) + x(*3)
x 1
Let u =(x+=) vexltts)
x.
y=utv
Differentiating both sides w.r. t. x.
dy d(tu+v)
dx dx
dy _ du + dv
dx dx dx

Ex5.5,7 teachoo.com
Differentiate the functions in, (logx)* + x!0&*
Let y = (log x)*+ x!08*
Let u = (logx)*, v= x!08*
y=utv
Differentiating both sides w.r. t. x.
dy d(u+v)
dx — dx
dy _ du + dv
dx dx dx

EX5.5, 8 teachoo.com
Differentiate the functions in, (sinx)*+ sin-t x
Let y = (sinx)* + sin-1-¥x
Letu =(sinx)* & v = sin-t-¥x
y=utov
Differentiating both sides w.r. t. x.
dy _d(utv)
dx — dx
dy _ du + dv
dx dx dx

teachoo.com

Ex 5.5,9
Differentiate the functions in, x°!"*+ (sin x)°S*
Let y= x5 * + (sin x)oS*
Letu = x5™* & vy = (sin x)°°S*
“y=utv
Differentiating both sides w.r. t. x.

dy _d(utv)

dx dx

dy du, dv

dx dx dx

teachoo.com
Ex 5.5, 10
. . . . +1
Differentiate the functions in, x* °8* + roar
eat Important-
Let y=x% 605% 4 ——
x- 1
xX COSX +1
letu =x &v ==
x1
“yu +v
Differentiating both sides w.r. t. x.
dy _d(utv)
dx dx
dy _ du + dv
dx dx dx

Ex 5.5, 11 teachoo.com
a
Differentiate the functions in, (x cosx)* +(x sinx) x
1
y=(xcosx)* + (sinx) =
1
Let u =(xcosx)*,v=(xsinx)*
yeoutyv
Differentiating both sides w. 7. t. x.
dy _d (u+v)
dx dx
dy _ du + dv
dx dx dx

Ex 5.5, 12 teachoo.com
+ dy + + x
Find x of the functionsin, x” +y*=1
xv ey*al
Let u =x” ,v=y*
Hence,
utv=l1
Differentiating both sides w.r. t.x.
dwvtu)_ dQ)
dx ~ dx
dv du wae .
—+—-=0 (Derivative of constant is 0}
dx dx

teachoo.com

Ex 5.5, 13
. ay oo x
Find a of the functionsin, y* =x”
Given,

yr ax?
Taking log both sides

log (y*) = log (x”)

x .log y = y.logx (As log(a”) = b.log a)
Differentiating both sides w.r. t. x.

d(x.logy) _ d(y.log x)
dx ~ dx

teachoo.com
Ex 5.5, 14
Find “ of the functions in, (cos x )” = (cos y )*
Given
(cos x)” = (cos y)*
Taking log both sides
log (cos x)” = log (cos y)*
y .log (cos x) = x. log(cos y) (As log(a”) = b.loga)
Differentiating both sides w. r. t. x.
d(y.log(cosx)) _ d(x. log(cos y))
dx ~ dx

teackoo.com
Ex 5.5, 15
Find ie of the functions in, xy = e@-¥)
Given
xy = e&-»)

Taking log both sides

log (xy) = loge@-»)

log (xy) =(x — y) loge (As log(a?) = b.log a)

logx + logy =(x —y) (1) (As loge = 1)

logx + logy =(x —y)

Ex 5.5, 16 teachoo.com
Find the derivative of the function given by f (x) = (1 +x) (1+ x7)
(1+x*) (1+ x8) and hence find f’(1).
Given
f@)=(4xn04+2x90 +2490 + x9)
Lety = (1+x)(14+x7)(1+ x71 + x8)
Taking log both sides
logy = log(l+x)+x7)1 +241 + x9)
(As log(ab) = log at log b)
logy = log (1 + x) + log(1 + x”) + log(1 + x*) + log (1 + x°)

Ex5.5, 17 teachoo.com
Differentiate (x? — 5x + 8) (x3 + 7x +9)
(ii) by expanding the product to obtain a single polynomial.
By Expanding the product to obtain a single polynomial .
y = (x? -5x +8) (x3 + 7x49)
y =x? (x3 + 7x4 9) —Sx(x3 + 7x49) 48 (x2 + 7x49)
y=x>+ 7x3 + 9x? — 5x4 — 35x2 — 45x 4 8x3 + 56x +472
y =x>— 5x44 15x39 — 26x27 4+ 11x + 72
Differentiating both sides w.r. t. x.
dy — d(x? — 5x44 15x3— 26x? + 11x +72)
dx dx
dy d(x) _ a(ex*) + d(15x7) _ d(26x?) + d(11x) + d(72)
dx dx dx dx dx dx dx

Ex5.5, 18 teachoo.com
If u, v and w are functions of x, then show that
d du dv dw
m (u. v.w)= vewtu. - wtu.v i
in two ways - first by repeated application of product rule,
second by logarithmic differentiation.
By product Rule
Let y = uvw
Differentiating both sides w.r. t.x.
dy _ d(uvw)
dx dx
dy _ d((uv) w)
dx dx

Ex 5.6, 1 teackoo
If x and y are connected parametrically by the equations
without eliminating the parameter, Find 2, x=2at?, y=at*
Here
ay
dy _ dt
dx ax
dt
_ dy _ dx
Calculating dt Calculating at
y =at* x =2at?
dy _ 4-1 dx _
dt = 4Aat dt =2 X 2at
= 4at3 = 4at

Ex 5.6, 2 teachoo.com
If x and y are connected parametrically by the equations without
eliminating the parameter, Find 2, x=acos@,y=bcosé
Here
dy
dy _ 40
dx ax
a0
Calculating = Calculating
do alcu ating 75
y =bcos@ x= acos@
dy _ a(bcos 0) dx _ d(acos 8)
do do do do
dy _ —e«j dx _ _
ae 7 Oh sin@) a 7a sin9)
dy _ : dx _ .
aa = bsin@ qa 7 asind

Ex 5.6, 3 teachoo.com
If x and y are connected parametrically by the equations without
eliminating the parameter, Find * x =sint,y = cos2t
Here,
dy
dy _ ae
ax
dt
. dy . dx
Calculating a Calculating a
dy _ d(cos 2t) dx] d(sin t)
dt” ~— at dt — dt
dy __ ax _
a sin 2t .2 ae 7 cost
d. :
“ =-2 sin2t
dt

Ex 5.6, 4 teachoo.com
If x and y are connected parametrically by the equations
without eliminating the parameter, Find, x= 4t,y = :
Here
dy
ay _ at
dx &
dt
ay 4
Calculating dt Calculating =
dy = <() dx_ d(at)
dt dt \t a = a
dy d {1
““=4—f- dx
dt 4k (7) —=4
dt
ay 4
att

Ex5.6,5 teachoo.com
If x and y are connected parametrically by the equations
_ 4d
without eliminating the parameter, Find ae
x = cos@- cos26,y = sin@- sin2@
Here
dy
dy _ ao
dx ak
ao
~ dy ax
Calculating = Calculating 7,
ay _ d(sin 6 - sin 26) dx _ d(cos@~cos 20)
do ae do” ao
ay = d(sin 0) _ d(sin 26) dx _ d(cos@) _ d{cos 26)
dé dé dé aos do
dy d
49 = COSA — cos26.2 =~ sind — (sin 20 .2
ay _ 0-2 20 dx . .
ae £08 cos ao = 7 Sin@ + 2sin26

Ex 5.6, 6 teachoo.com
If x and y are connected parametrically by the equations
without eliminating the parameter, Find 2,
x = a(@-sin@),y = a(1 + cos@)
Here
dy
dy _ do
dx ax
doe

Ex 5.6, 7 teachoo.com
If x and y are connected parametrically by the equations
without eliminating the parameter, Find *
_ _sinst _ cost
x= vcos 2t” y= ¥cos 2
Here,
ay
ay _ ae
ax ax
at

teachoo.com
Ex 5.6, 8
If x and y are connected parametrically by the equations
without eliminating the parameter, Find ae
x= a(cost + log tan=), =asint
Here
dy
ay _ de
dx a
dt

Ex 5.6, 9 teachoo.com
If x and y are connected parametrically by the equations
without eliminating the parameter, Find,
x=a sec 6,y = btand
Here
dy
dy _ dé
dx ax
dé
. ay . dx
Calculating 70 Calculating qo
dy _ d(btan 6) dx _ d(asec 8)
de°—sé«O ao a0
ay _ b d(tan @) ax _ a(sec 0)
ado” a0 ao" a6
ay _ 2 ax _
ap = D-Sec 0 ap 7 2 ([email protected]®)

teachoo.com
Ex 5.6, 10
If x and y are connected parametrically by the equations
without eliminating the parameter, Find a
x = a(cos@ + @sin@),y = a (sind - 6cos@)
Here
dy
ay _ ae
dx ax
ao

€x 5.6, 11 teachoo.com
-6,
—— = a
fx =yasr tt, y=vyaes"t, show that > =-2
Here
dy
ay _ ae
dx ax
dt

teachoo

Ex 5.7,1
Find the second order derivatives of the function x? + 3x +2
Let y =x? 43x+2
Differentiating w.r.t.x

dy _ d(x? +3x+2)

dx dx

2

dx ax ax ax

d

i 2x4+340

dx

dy

a 2x +3

teachoo.com

Ex 5.7, 2
Find the second order derivatives of the function x2°
Let y = x2°
Differentiating w.r.t.x

dy _ d(x?°)

dx dx

PY = 29x20-1

dx

d

> = 20x"

dx
Again Differentiating w.r.t.x

d (2) _ a (20x1)

dx \ax} ~ ax

teachoo.com

Ex 5.7, 3
Find the second order derivatives of the function x. cos x
Let y =x. cos x
Differentiating w.r.t.x .

dy _ d(x. cos x)

dx dx
Using Product Rule
As (uv)’= uv + vu

dy _ d(x) d(cos x)

de dy (OSH +

dy _ _

ae = COS% +(-—sinx).x

Ex5.7, 4 teachoo.com
Find the second order derivatives of the function log x
Let y = log x
Differentiating w.r.t.x .

dy _ d(log x)

dx ~ dx

dy oi

dx x
Again Differentiating w.r.t.x

ila) =a)

dx \dx} ~ dx\x

teachoo.com

Ex 5.7,5
Find the second order derivatives of the function x° log x
Let y=x? logx
Differentiating w.r.t.x .

dy _ d(x* logx)

dx dx
using product rule in x?
log x.
As (uv)’= wv + vu
where u =x? & v=log x

dy — a(x?) d(log x)

==.) +

dx dx 98% dx *

Ex 5.7, 6 teachoo.com
Find the second order derivatives of the function e* sin5x
Let y= e* sin 5x
Differentiating w.r.t.x .

dy _ d(e* sin 5x)

dx” dx
using product rule in e* sin 5x
As (uv)’= uv + vu

d d(e~ d(sin5 x

o. ae") .sin 5x+ asin 5%) ax

dx dx dx

d d(5x

& =e* .sin 5x + cos 5x 46%) e*

dx dx

d

& = e*. sin 5x+5.e%. cos 5x

dx

EX5.7,7 teachoo.com
Find the second order derivatives of the function e® cos 3x
Lety = e* cos3x
Differentiating w.r.t.x.
dy _ de® cos 3x)
dx dx
Using product rule in e® cos 3x.
As (uv)’= wv + vu
where u = e* & v=cos 3x
dy _ a(e®) d(cos3x)
— = —~— .cos 3x + ———— .e™*
dx dx dx
dy 6x 46%) . d(x) 6x
—= _—. +(- —_.
x 78 am oS 3x + (—sin 3x) me

Ex 5.7,8 teachoo.com
Find the second order derivatives of the function tan7+ x
Let y= tan-tx
Differentiating w.r.t.x .

dy _ d(tan7+ x)

dx dx

dy _ 1

dx 1+x2
Again Differentiating w.r.t.x

i (as) > a (aa)

dx\dx/ dx \1+x?

dy d ( 1 )

dx? ~ dx \1+x2

Ex5.7,9 teachoo.com
Find the second order derivatives of the function log (log x)
Let y=log (log x)
Differentiating w.r.t.x .

dy _ d(log dog x))

dx — dx

dy 1 d(log x)

dx logx’ dx

dy _ 1 1

dx logx ‘x

dy _ 1

dx — x.logx

teachoo.com

Ex 5.7, 10
Find the second order derivatives of the function sin (log x)
Lety= sin (log x)
Differentiating w.r.t.x .

dy _ d( sin (log x))

dx — dx

dy _ d(log x)

a 7 cos(log x) a

dy _ 1

ms cos(logx) 5

dy _ cos(log x)

dx x

Ex 5.7, 11 teachoo.com
. ay
Ify =5 cosx —3sinx ,prove that me te 0
y=5 cosx —3sinx
Differentiating w.r.t.x
dy _ a(S cos x—3 sin x)
dx dx
dy _d(Scosx) dQ@sin~x)
dx dx dx
d
& = -Ssinx -3cosx
dx
Again Differentiating w.r.t.x
d (2) _ d(—5sin x — 3cos x)
dx\dx} ~ dx

teachoo.com
Ex 5.7, 12
-1 . ey,

If y=cos~* x, Find qa interms of y alone.
Let y=cos-tx
Differentiating w.r.t.x

dy _ d(cos7* x)

dx — ax

dy = =1 A a(cos-tx) — =1

dx V1—x2 rn
Again Differentiating w.r.t.x

a (2) 4 (=)

dx\dx} ~ dx\V1—x2

Ex5.7, 13 teachoo.com
If y=3 cos (logx) + 4 sin (logx),
show that x?y, +xy,+y=0
y =3cos (logx) + 4 sin (log x)
Differentiating w.r.t.x
dy _ 3 1 1
an =-3 sin (log x) x= +4 cos (log x} x=
oy 3 sin (log x} + 4 cos (log x}
X 3 = 73 sin (log x cos (log
Differentiating w.r.t.x
dy)" , ,
(x ~) = (-3 sin (log x))’ + (4 cos (log x})

Ex 5.7, 14 teachoo.com
TF,
a a.
If y =Ae™* + Be™, show that -(m+n) a +mny =0
y = Ae™* +Be™
Differentiating w.r.t.x
dy _ a(Ae™ + Be?)
dx dx
dy _ ad(Ae™*) + d(Be™™)
dx dx dx
dy =A.em d(mx) +B.e™ d(nx)
dx . “dx . dx
d
= Ale™,) m+B.e™ n
dx
d
<= Ame™ +Bne™
dx

Ex 5.7, 15 teachoo.com
If y = 500e7*+ 600e~”*, show that < = 49
y = 500e7*+ 600e-”*
Differentiating w.r.t.x
dx” dx
dy _ d(500e7*) + d (600e~7*)
ax ~ dx ax
dy _ d(e”*) d(e-7*)
7, = 500——— _ + 600 ——
dy _ x A(7%) ay 4 (-72)
Ig 7 300.€%. = + 600.e°7* . —

Ex 5.7, 16 (Method 1) teachoo.com
y - &y _ (ay?
Ife” (x+ 1) = 1, showthat = = (2)
We need to show that
ay (2y
dx? ~~ \ax
ex+t)H=1
Differentiating w.r.t.x
a(e¥(x+1)) _ aq)
dx "ax
a(eY (x+1)) _ 0
dx ~

teachoo.com

Ex 5.7, 17 (Method 1)
If y = (tan-1 x)”, show that (x? + 1)? y,+ 2x (x? +1) y,=2
We have

y = (tan? x)?
Differentiating w.r.t.x

"=2tantxx a

ye 1+ x4

(14+x7)y’ =2tantx
Again differentiating w. r. t.x

[ (1+ x*)I =2x 5

y 1+x? 1+x?

b+ =

y 14+ x?

Example 1 teachoo.com
Check the continuity of the function f given by f (x) = 2x + 3 at x= 1.
f (x) is continuous at x = 1
if lim f(x) = fq)
xXOL
L.H.S R.H.S
lim f(x) f@)
x?
= 2x1+3
= lim (2x + 3)
sot = 2+3
=2x1+3
=5
=2+3
=5
Since, L.H.S = R.H.S
-. Function is continuous.

Example 2 teachoo.com
Examine whether the function f given by f (x) = x? is
continuous atx = 0
f (x) is continuous at x = 0
if lim f(x) = f(0)
x70
L.H.S R.H.S
i f(0)
lim f(x)
= (0)?
= lim x?
x70 =0
Putting x = 0
= (0)?
=0
Since LHS = RHS
Hence, f(x) is continuous at x = 0

Example 3 teachoo.com
Discuss the continuity of the function f given by
f(x) =|x| atx = 0.
f@) = |x|
_ j-xifx<0

f) = {i >0
f is continuous at x = 0

if LH.L=RH.L= f(0)

ie. lim f@)= lim, fO) = FO)
Finding LHL and RHL

Example 4 teachoo.com
Show that the function f given by
_ f2+3, ifx #0
roo = {fF ifx=0
is not continuous at x = 0.
f(x) is continuous at x = 0
if LH.L= RHL= £0)
If lim f Ge) = lim, f%) = f(0)
Finding LHL and RHL

Example 5 teachoo.com
Check the points where the constant function f (x) = k is
continuous.
Given f(x) = k (where k is any constant)
To check continuity of f (x),
We check it’s if it is continuous at any point x = c
Let c be any real number
fis continuous atx =c

if lim f(@) = f(c)

x-c

Example 6 feachoo.com
Prove that the identity function on real numbers given by f
(x) = x is continuous at every real number.
Given f(x) =x
To check continuity of f (x),
We check it’s if it is continuous at any point x = c
Let c be any real number
fis continuous atx =c

if lim f(@) = f(c)

x-c

Example 7 teachoo.com
Is the function defined by f (x) = |x|, a continuous function?
fl) = |x|
_ 4-x, x<0
~ )x, x20
Since we need to find continuity at of the function
We check continuity for different values of x
* Whenx=0
* Whenx<0
* Whenx>0

Example 8 teachoo.com
Discuss the continuity of the function f given by
f@=xeP4+xH-1.
Given

f(x) = 84+ x7 - 1.
To check continuity of f(x),
We check it’s if it is continuous at any point x = c
Let c be any real number
fis continuousatx =c

if lim f(x) = f(c)

x-c

Example 9 teachoo.com
Discuss the continuity of the function f defined by
1
f@ = yk # 0.
+ 1
Given f (x) = x
Atx = 0
1
f (0) =+
= 00
Hence, f(x) is not defined at x = 0
By definition, f (x) = = x #0.
So, we check for continuity at all points except 0.

Example 10 teachoo.com
Discuss the continuity of the function f defined by
x+2,ifxe1
roy = {785 ifx>1
_ (xt+2,ifx<s1
ro = {E93 if x>1

Since we need to find continuity at of the function
We check continuity for different values of x

« Whenx=1

« Whenx<1

« Whenx>1

Example 11 teachoo.com
Find all the points of discontinuity of the function f defined by
x+2 ,ifx<1
f@)=<40 »ifx=1
x-2 ,if x>1
x+2 ,ifx<1
f@)= 40 »ifx=1
x—-2 ,if x>1
Since we need to find continuity at of the function
We check continuity for different values of x
* Whenx=1
* Whenx<1
* Whenx>1

Example 12 teachoo.com
Discuss the continuity of the function defined by
x+2, ifx<0
roy= {NF if x>0
_f x+2, ifx<0
ro = {255 if x>0
Here, function is not defined for x = 0
So, we do not check continuity there
We check continuity for different values of x
« Whenx<0
« Whenx>0

teachoo.com
Example 13
Discuss the continuity of the function f given by
_{ x, ifx20
Fo) =| x, if x<0
_{ x, ifx20
f= | x, if x<0
Since we need to find continuity at of the function
We check continuity for different values of x
* Whenx=0
* Whenx<0
* Whenx>0

teachoo.com

Example 14

Show that every polynomial function is continuous
Let f(x) =agta,xtayx? + .. +a,x" neZz
be a polynomial function
Since Polynomial function is valid for every real number
We prove continuity of Polynomial Function at any point c
Let c be any real number
f(x) is continuous at x =c

iflim f(x) = f(c)

x-c

. teachoo.com
Example 15 (Introduction)
Find all the points of discontinuity of the greatest integer
function defined by f (x) = [x], where [x] denotes the greatest
integer less than or equal to x
Greatest Integer Function [x]
Going by same Concept
Value of
Value of x | Greatest Integer Value of
bd Value of x | Greatest Integer
3 3 Ix]
c c
3.1 3
ct=cth c
3.9999 3
ce -=c-h c-1
4 4
4.0001 4

Example 16 teachoo.com
Prove that every rational function is continuous.
Every Rational function f(x) is of the form
p(x)
x) = ——
[0 = Tm
where q(x) #0 & p, q are polynomial functions
Since p(x) & q(x) are polynomials,
and we know that every polynomial function is continuous.
Therefore, p(x) & q(x) both continuous

teachoo.com

Example 17
Discuss the continuity of sine function.
Let f(x) = sinx
Let’s check continuity of f(x) at any real number
Let c be any real number.
We know that
A function is continuous at x =c

if LH.L = RH.L= f(e)

ie. lim f(x) = lim, f(x) = fc)

Koc xX?C

Example 18 teackoo.com
Prove that the function defined by f (x) = tan x is a continuous
function.
Let f(x) =tanx
sin x
f@) ~ cos x
Here,
f (x) is defined for all real number except cos x =0
i.e. for all x except x = (2m + 1) 5
Let p(x) = sinx & q(x) = cosx
We know that sin x & cos x is continuous for all real numbers.
Therefore, p(x) & q(x) is continuous.

Example 19 teachoo.com
Show that the function defined by f (x) = sin (x?) is a continuous
function.
Given
f(x) =sin(x?)
Let g(x) = sinx & A(x) =x?
Now,
(g 0 h)(x)
= g(h(x))
= g(x’)
= sin(x?)
= f(x)

Example 20 teachoo.com
Show that the function f defined by
f(x)=|1— x + |x ||, wherex is any real number is a continuous
Given f(x) = |1 —xt+ Ix||
Let g(x)=1—x+ |x|
& h(x) = |x|
Then,
hog(x)
= h(g(x))
=A —x + |x])
= |1 —x+ Ix]|
= f(x)

teachoo.com

Example 21
Find the derivative of the function given by f (x) = sin(x?).
Let y= sin(x?)
We need to find derivative of y, w.r.t.x
_. dy — d(sin x?)
re. dx dx

= cosy? , 22

= COSX? .

= cosx? . (2x?71)

=cosx? (2x)

=2x.cos x?

teachoo.com
Example 22
ay. _
Find ax ifx — yen.
(x-y)=1
Differentiating both sides w.rt x
d(x-y)_ dn
dx ~ dx
dx dy _ .
i an 0 (As 7 is constant)
d
1-2=0
dx
dy
dx 1

Example 23 teachoo.com
Find = , if y+siny=cosx
aitytsiny=
ytsiny =cos x
Differentiating both sides by x
dy + a(siny) _ d(cos x)
dx dx —s dx
d d(sin .
ay , Hsin ¥) _ _ gin y
dx dx
d d(sin d .
a+ asin y) 2 =-sinx
dx dy dx
d d .
+ cosy — =-sinx
dx dx

Example 24 teackoo.com
Find the derivative of f given by f (x) = sin-' x assuming it exists.
f@)=sin 1x
lety = sin"1x
siny=x
x=siny
Differentiating both sides w.r.t.x
dx_d (sin y)
dx dx
d({siny) dy
t= OO xe
dx dy

teachoo.co
Example 25 "
: I

Is it true thatx =e °” forall real x?

x=e log x
Forx=0

O= elog 0
But log 0 is not defined.
Hence, the equation is not defined for x = 0

Forx <0
x=e log x

But log x is not defined for negative numbers
Hence, equation is not defined for x <0

teachoo.com

Example 26
Differentiate the following w.rt. x:
{i)e*
Llety =e*
Differentiating both sides w. r. t.x

dy _ d(e*)

dx dx

dy =e d(—x)

dx “dx

dy x

a 7 e (-1).

dy -x

dx e

Example 26 Differentiate the following w.r.t. x: (ii) sin⁡(log⁡𝑥), 𝑥 > 0
Let 𝑦 =sin⁡(log⁡𝑥)

Example 26 Differentiate the following w.r.t. x: (iii) 〖𝑐𝑜𝑠〗^(−1) "(ex)" Let 𝑦 = 〖𝑐𝑜𝑠〗^(−1) "(ex)"

Example 26 Differentiate the following w.r.t. x: (iv) 𝑒𝑐𝑜𝑠 𝑥Let 𝑦 = 𝑒^cos⁡𝑥

Example 27 teachoo.com
_ 2
Differentiate jee w.r.t.x.
3x*+4x4+5
(x — 3) (x2 +4)
Let y= =e
3Bx°+4x4+5
Taking log on both sides
_ [a-G*+4)
logy =log 3x74 4x45
2 1
_ (x — 3) (x? + 2)
logy = log ( Bx7+4x4+5
aNog SIDE Using loga® = bloga
logy =7108 Garazas ‘(Using log Ba)

Example 28 teachoo.com
Differentiate a* w.r.t.x, where a is a positive constant.
Let y =a*
Taking log on both sides

logy = loga*

logy =xloga (log a? = bloga)
Differentiating both sides w.r.t.x

ddogy) _4

a ae (xlog a)

ddogy) _ ax

a. loga (=)

ddogy) _ loga

dx ='0g

Example 29 teachoo.com
Differentiate xS™*, x > O w.r.tx.
Let y= x5in*
Taking log both sides
logy = log x5in *
logy=sinx.logx (loga® =bloga)
Differentiating w.r.t.x
d(logy)_d |.
ax dx (sin x log x)
By product Rule
(uv)’=u’v + vu
where u = sin x & v= log x

teachoo.
Example 30 CACHOO.LOM
Find &, if y* 42% +x% =a?
dx’ .
Let u=y*, vex” &we= x*
Now,
ut+v+we=a
Differentiating w.r.t.x
d(u+viw) _ aca?)
dx "dx
au), a) | dw) _ (As a” is constant) (1)
dx dx dx
We will calculate derivative of u, v & w separately .

Example 31 teachoo.com
. dy . .
Find a if x = acos6,y = asin®.
dx
Here
dy
dy _ a6
dx ax
ao
: dy . dx
Calculating 70 Calculating aa
y =asin®8 y =acosd
dy _ d(asin @) dy _ d(acos 6)
do” ~— do” do
d d .
“= acos@ “= -asin@
do do

Example 32 teachoo.com
. dy. _ 2 _
Find —, ifx = at?,y = 2at.
dx
Here
dy
dy _ at
dx ax
dt
. ay . dx
Calculating at Calculating 77
y = 2at x =at?
dy _ d(2at) dx _ d(at’)
dt dt dt ~ dt
dy 94 a dx a)
ae 24a a ae
dy dx
at =2Za at =2at

Example 33 teachoo.com
Find =, if x = a(@+sin@),y = a(1- cos@)
Here
ay
ay _ a6
dx ax
ae
_ ay da
Calculating de Calculating a
y =a(1- cos@) x =a(O+sin8@)
dy _ a(a (1~ cos 6)) dx d(a(@+sin@))
d@ d@ ae = de
dy . .
— =a(0 —(-siné@) ax _ | (ao, alsin 8)
ae ( ) aa 72 (aot do )
dy _ . d
qq 7 @(sin8) 7a +0088)

Example 34 (Method 1) teachoo.com
2 2 2
Find =, if xe+y3 =a3,
dx
2 2 2
x3+ y3 = a3
Differentiating w.rt. x
2 2 2
d(x3) 4 d(y3) _ d(a3)
dx dx dx
2 2
20 =-141 d(y3) d
22-14 403) ay 9
3 dx dy
1 2
2 = dvy3 da
2,3, a0), ay _ 9
3 dy dx
2. 2.24. ay
—-x3 +-y3 x—=0
3 3 ¥ dx

Example 35 teachoo.com
Find vy if y = x3 +tanx
ind Saif y = .
y = x3+tanx
Differentiating w.r.t.x
dy _ d(x3+tanx)
dx ax
dy _ d(x) + d(tan x)
dx ax ax
d
23x? + sec?x
dx
Again Differentiating w.r.t.x
a?y _ da (3x? tsec? x)
ax2 — ax

Example 36 teachoo.com
. ay
lfy = Asinx + Bcosx, then prove that ae tYe 0.
y = Asinx +Bcosx
Differentiating w.r.t.x
dy _— d(Asinx+Bcosx)
dx — dx
dy _ d(Asin x) + d(B cos x)
dx dx dx
d. d(sin x d(cos x
dy _, dsinx) |, d(cosx)
dx dx dx
d
& =Acosx +B (- sin x}
dx
dy .
— =Acosx -Bsinx
dx

Example 37 teachoo.com
ay ay
= 2K 3K —- ~-5— =

If y = 3e* + 2e**, prove that a Sox + 6y =0.,
Given,

y = 3e% + 263
Differentiating w.r.t.x

dy _ d(e*+ 2e*)

dx dx

dy de”) sae)

dx dx dx

dy _ ox d(2x) yy AGBx)

am” 3.e a +2.e3

d

& = 3.6.24 2.¢3%.3

dx

Example 38 (Method 1) teachoo.com
—_ ay dy
= 1 _ y2y OY _y
If y= sin~*x, show that (1 - x7) a ax oO.
We have
y=sin tx
Differentiating w.r.t.x
dy _ a(sin™*x)
dx — dx
dy - 1 (A a{sin"tx) 4 )
dx V1—x2 8 ie ~ Vix?
V(1—2?) y'=4
Squaring both sides

Example 39 teachoo.com
Differentiate w.rt. x, the following function:
. 1
(i) V38x+2 + Ea
L
Let y= V3x + 2+
Differentiating w.r.t.x
a vax ++ ——)
dy _ vax? +4
dx dx
wa) . (Ee)
dy _ a(v¥3x+2) + 2x2 + 4
dx dx dx
a1
dy _ d(v3x+2) + d(2x? +4)2
dx dx dx

Example 39 Differentiate w.r.t. x, the following function: (ii) log7 (log x) y = log7 (log x)

Example 40 (Method 1) teachoocom
Differentiate the following w.r.t.x.
(i) cos~* (sin x)
Let f(x) = cos“ (sin x)
f (x) = cos™* (cos G - x)) (As sin@ =cos G - x))
f@)=5-*
Differentiating w.r.t.x
a G) d(x) (As 1 27 gn is constant)
, =—2/ _ oN
f ) ~ dx dx “ ,
P@)=0-1
f@=-t

Example 40 (Method 1) Differentiate the following w.r.t. x. (ii) tan −1 (sin⁡𝑥/( 1 +〖 cos〗⁡〖𝑥 〗 ))
Let 𝑓(𝑥) = tan −1 (𝒔𝒊𝒏⁡𝒙/( 1 +〖 𝒄𝒐𝒔〗⁡〖𝒙 〗 ))

Example 40 Differentiate the following w.r.t. x. (iii) sin^(−1) ((2^( 𝑥+1) )/( 1 +〖 4 〗^𝑥 ))
Let 𝑓(𝑥) = sin^(−1) ((2^( 𝑥+1) )/( 1 +〖 4 〗^𝑥 ))
𝑓(𝑥) = sin^(−1) ((2^( 𝑥). 2)/( 1 + (2^𝑥 )^2 ))

Example 41 teachoo.com
Find f‘(x) if f (x) = (sinx)S™* foralld<x<n.
Let y = (sinx)si"*
Taking log on both sides
log y = log (sin asin *)
log y = sinx . log (sin x) (As log(a”) = b. log a)
Differentiating both sides w. r. t. x
d(logy) _ a(sin x. log (sin x))
ax ~ ax
d(log y) (2) _ d(sin x. log (sin x))
dx dy ~ dx

Example 42 teachoo.com
sgt . dy
For a positive constant a find ae’ where
1 2
y =a'ti,and x = (t++)
Here
dy
dy _ ae
dx &
dt

teachoo.com
Example 43
Differentiate sin?x w.r.t.e°°S*:
Let u =sin’x & v =e°S*
We need to differentiate u wrt. v.
. du
ie, —
dv
Here,
du
du _ ik
dy ww
dx

Example 22 teachoo.com
Find the derivative of tan (2x + 3).
Let y = tan (2x + 3}
We need to find derivative of y,
_ dy _ dtan(2x+3)
ie.
= sec?(2x + 3} x “o> (As (tan x)’ = sec? x)
oa
= sec? (2x +3) x2
= 2 sec? (2x +3)

Example 23 teachoo.com
Differentiate sin (cos (x?)) with respect to x.
Let y= sin (cos x?)
We need to find derivative of y w.r.t.x
ie. y’ = (sin (cosx7))
dy _ d(sin (cos x?))}
dx dx
_ 2 d(cos x?)
= cos (cos x“) a
2
= cos (cos x”). (— sinx?) . ate?)
dx
= cos (cos x“). (—sinx?) . 2x
=- 2x sin x”. cos (cos x”)

\

teachoo.com

Example 42
Verify Rolle’s theorem for the function y = x? + 2,a@=-—2 and b= 2.
y=x2+2,q=-2andb=2

Conditions of Rolle’s theorem
Let f(x) =x? +42 1. f (x) is continuous at (a,b)
Rolle’s theorem is satisfied if 2. f (x) is differentiable at (a, b)

3% f(a)= Ff)

Condition 1

if all 3 conditions satisfied then
Since f (x) is a polynomial,

there exist some cin (a,b)
it is continuous
rs inuou such that f'(c) =0
«. f (x) is continuous at (—2 , 2)

Example 43 teachoo.com
Verify Mean Value Theorem for the function f(x) = x? in the
interval (2, 4].
f(x) = x? in interval (2, 4].
Checking conditions for Mean value Theorem

Conditions of Mean value theorem
Condition 1

1. f(x) is continuous at (a, b)

Since f(x) is polynomial . 2. f(x) is differentiable at (a, b)
it is continuous
“. f(x) is continuousat (2, 4) | if both conditions satisfied, then

there exist some cin (a,b)

such that f'(c) =LO-L@

Example 44 Differentiate w.r.t. x, the following function: (ii) 𝑒^(sec^2⁡𝑥 ) + 3cos^(–1) 𝑥 Let y = 𝑒^(sec^2⁡𝑥 ) + 3cos^(–1) 𝑥
Differentiating 𝑤.𝑟.𝑡.𝑥
𝑑𝑦/𝑑𝑥 = 𝑑(𝑒^(sec^2⁡𝑥 )+ 3cos^(–1) 𝑥" " )/𝑑𝑥
𝑑𝑦/𝑑𝑥 = 𝑑(𝑒^(sec^2⁡𝑥 ) )/𝑑𝑥 + 𝒅(𝟑〖𝒄𝒐𝒔〗^(–𝟏) 𝒙)/𝒅𝒙
𝑑𝑦/𝑑𝑥 = 𝑒^(sec^2⁡𝑥 ) 𝑑(sec^2⁡𝑥 )/𝑑𝑥 + 3. ((−𝟏)/√(𝟏 −𝒙^𝟐 ))
𝑑𝑦/𝑑𝑥 = 𝑒^(sec^2⁡𝑥 ). 2 sec 𝑥 . 𝒅(〖𝒔𝒆𝒄 〗⁡𝒙 )/𝒅𝒙 − 3/√(1 −𝑥^2 )
"As" 𝑑(𝑒^𝑥 )/𝑑𝑥=𝑒^𝑥
& 𝑑(〖𝑐𝑜𝑠〗^(−1)⁡𝑥 )/𝑑𝑥=(−1)/√(1 −〖 𝑥〗^2 )
𝑑𝑦/𝑑𝑥 = 𝑒^(sec^2⁡𝑥 ). 2 sec 𝑥 . 𝒔𝒆𝒄⁡𝒙 .𝒕𝒂𝒏⁡𝒙 − 3/√(1 − 𝑥^2 )
𝒅𝒚/𝒅𝒙 = 〖𝟐 𝒆〗^(〖𝒔𝒆𝒄〗^𝟐⁡𝒙 ). 〖𝒔𝒆𝒄〗^𝟐 𝒙 .𝒕𝒂𝒏⁡𝒙 − 𝟑/√(𝟏 − 𝒙^𝟐 )

Mise 1 teachoo.com
Differentiate w.rt. x the function,
(3x2 - 9x + 5)?
Let y = (3x2- 9x + 5)?
Differentiating w. r. £. x.
dy — a(3x?-9xt 5)?
dx dx
d(3x?-9x +5)
= 2_ 9-1
9x? - 9x + 5) . ax
d(3x? d(9. a(S
= 9(3x2 - 9x + 5° (2 — d(x) = )
dx dx dx
= 9(3x2- 9x + 5)®. (6x -9+0)
= 9(3x2- 9x + 5)®. (6x — 9)

Misc 2 teachoo.com
Differentiate w.r.t.x the function
sin? x + cos°x
Let y=sin? x + cos® x
Differentiating w. r. t. x.
a(( ingx) + 6
dy sin? x) + (cos® x }
dx dx
d( (sin? x d((cos® x
dx dx
d(sin x d(cos x.
= 3 sin’x asin) + 6 cosSx , ees ™)
dx dx
=3sin?x .cosx + 6 cos°x. (—sinx)

Misc 3 teachoo.com
Differentiate w.r.t.x the function, (5x)?°s 2*
Let y= (5x)300s 2x
Taking log on both sides
logy =log (5x)3°°s 2%
log y=3 cos 2x .log 5x (As log(a’) = bloga)
Differentiating both sides w.r. t. x
d(log y) _ dG cos 2x. log 5x)
dx ~ dx
d(log y) (2) _ a3 cos 2x. log 5x)
dx dy ~ dx

Misc 4 teachoo.com
Differentiate w. r.t. x the function,
sin (x ¥x),0 <x <1
Let y =sin™* (x yx)
1
y=sin' (x. x2)
— ein t ppt t%
y=sin™ (x "2)
3
y= sin} (x2)
Differentiating w. r. t. x
3
dy _ a{sin-2 (x2))
dx dx
d oe
y 1 d(x)2 4
“es , x (As d{sin x) 1 )
dx (2) ax dx Vi-x?
1-\x2.

. teachoo.co
Misc 5 "
Differenti he function, 2 ,- 2 2
ifferentiate w.7r.t.x the unction, 7 <x <
cos"
Let y = ——=
y V2X47
Differentiating both sides w.r. t. x
-1%
dy _ a [cos*>
dx dx V2X+7
Using Quotient rule
As (“) = ulv au
v v
where u = cos*> &vev2x+7

. teachoo.co
Mise 6 (Method 1) "
Differentiate w.rt. x the function,

pol vil+sinx +v1-sinx Q< <t
co ——_—__— ], xe

vit+sinx —v1—-sinx 2
Let y = cot7? Vv¥1l+sinx+V1- sinx
y= vitsinx—-y1-—sinx
Rationalizing the sum
fa (V1 + sin x + ¥1—sinx) y (iL + sinx + vi— sin x)
= co a ——
y (Vi+sinx-Vi-sinx) (Vi+tsinx + V1 -sinx)
2
t (V1 + sin x + V1 — sin x)

=co ee

y (VI + sinx - ¥1-sinx) (VI+sinx++¥1-sinx)

teachoo.com
Misc 7
Differentiate w.rt. x the function,
(log x) 8 >1
log x
Let y = (log x)
Taking log both sides
logy =log (Cog x) °8*)
log y =logx. log (log x) (As log(a”) = bloga)
Differentiating both sides w.r. t. x.
d(logy) _ dog x.log (log x))
ax ~ dx

Mise 8 teachoo.com

Differentiate w.7r.t.x the function,

cos (a cos x + b sin x), for some constant a and b.

Let y=cos (a cosx + bsinx)

Differentiating w.r.t.x.

dy _ 4(cos((acos x+b sin x))

ax dx

wy —sinx (a cosx + b sinx) _ Ma cosx+b sin)

dx dx
-_« : d(cos x) d(sin 2)
=—sinx(acosx+bsinx). (a. 2 +b a
=—sinx(acosx+bsinx). (a(- sinx) + b (cos x))

teachoo.co

Misc 9 "
Differentiate w.rt. x the function,
(sin x — cos x)6Sin ¥—C08 2) . <x< ae
Let y = (sinx — cos x)(sin *-c0s x)
Taking log on both sides

logy =log (sin x — cos x) in *—cos *)

log y = (sin x — cos x). log (sin x — cos x)
Differentiating both sides w.r. t. x.

d(logy) _ a((sin x — cos x). log(sin x — cos x))

dx ~ dx

Misc 10 teachoo.com
Differentiate w.rt. x the function,
x* + x* + a* + a*, forsome fixeda > Oandx> 0
Lety = x* + x7 + a*+ a?
Andletu = x*, v=x*, w=a*
Now,
y=ut+vtiwtast

Differentiating both sides w.r. t. x.

dy d(u+v+wta’)

dx dx

dy dtu) , dv) , dw) d(a*)

dx dx te bax + dx

Misc 11 teachoo.com
Differentiate w.rt. x the function,
x’-3 4 (¢ —3)" ,forx > 3
Let y =x7°-3 + (x — 3)?”
Andlet u=x* ~?,v=(x-3)*
Now,

y=ut+v
Differentiating both sides w.r. t. x.

dy _ d(tut+v)

dx dx

dy _ du + dv

dx dx dx

teachoo.com
Misc 12
Find 2 if y = 12 (1-cost),x=10 (t- sint),-—F <x< 5
Here,
dy
dy _ dt
dx dx
dt

Misc 13 teachoo.com
Find =, if y=sin-'x+sin'tV1—2x2,-1<x%<1
y=sinixtsintv1i-x*, -1<x<1
Puttingx = sin®@

y = sin-1 (sin@) + sin“! V1 — sin2@

y = 6+ sin"! Vcos26

y =@+sin™ (cos @)

y=@+sin* (sin G - 6)) (As cos @ = sin §-8))

we
y=9+(5-9)

Misc 14 teachoo.com
| Vf - _ dy -1
lfxJltytyvit+xe= 0, for 1<x<1, prove that — - Gay
xJjlt+ytyv1+x=0
x f/l+y =-yv1+x
Squaring both sides
2 2
(xf/1+y) = (-yv1+x)
2 2
x? (J/1+y ) = Gy? (WI+x)
2 = y2
x(1+y) = y* +x)
x? +xty = y? + yx

Misc 15 teachoo.com
If (x- a)? + (y- b)? = c2,forsomec > 0, prove that
3
dy 272
[+ @) ;
—qy ‘Isa constant independent of a and b.
x2
. . dy
First we will calculate —
dx
(x- a)? + (y- bY = ¢?
Differentiating w.r.t. x.
a((x-a)?+(y-b)?) ac?)
dx ~ dx
a{ (x - a)?) a((y - b)?)
——_——— + —~-0
dx dx

Misc 16 teachoo.com
If cosy =x cos(a + y), withcosa # + 1, prove that
dy _ cos?(at+y)
dx sina
Given
cosy = x cos(a + y)
cosy _
cos(a+ y) =x
_ cosy
x* cos(at+ y)
Differentiating w.r.t.x.
d(x) _ =( cosy )
dx dx cos(a + y)

teachoo.co
Misc 17 aenoo.com
2
Ifx =a(cost + tsint) andy =a (sint - t cos t), Find <=
dy
We need to find —>
ax?
d
First we find “YY
dx
Here,
dy
ay _ dt
dx ax
dt

Misc 18 teachoo.com
if f (x) = [x|?, show that f ”(x) exists for all real x and find it.
We know that
l=} % 228
—x x<0
Therefore,
ype} @ ,x*20
f@=ll hex <0
{x ,x20
—x3 »x<0

Misc 19 teachoo.com
Using the fact that sin(A + B) =sinAcosB +cosAsinB
and the differentiation, obtain the sum formula for cosines.
Given
sin(A + B) = sinAcosB + cosA sinB
Consider A & B are function of x
Differentiating both side w.r.t.x.
d(sin(A+ B)) _ d(sinAcosB +cosA sin B)
dx ~ dx
d(sin(A+ B)) _ d(sinA.cosB) 4 d(cosA.sinB)
dx ~ dx dx
cos (A +B) ; d(A+ B) = d(sin A.cosB) 4 d(cos A.sinB)
dx dx dx

Misc 20 teachoo.com
Does there exist a function which is continuous everywhere but
not differentiable at exactly two points? Justify your answer.
Consider the function
f@) = Ix] +|x—-1]
f is continuous everywhere , but it is not differentiable at
x=O0&x=1
-x-(x-1) x<0
f@)=4x-@-1 O<x<1
x+(x-1) x21
—2x+1 x<0
= 1 O<x<1
2x—-1 x21

Misc 21 (Method 1) feachoo.com
f@) g&) kh) ay [FO gD
lfy=|l om n |,prove that =| 1m n
a b c a b c
a» [FO I@) WO)
Here = = 1 Mm n
ax
a b c
Expanding determinant
dy _\¢ mn os Lon 1 Lom
Za, Ae’ @ll, Ta+ reo, 7]
° = f'(x) (me — bn) — g’(n) (lc — an) + h'(n) (lb — am)
° =(me — bn) f'(x) — (lc — an)g'(x) +(lb — am) h'(x)

Misc 22 teachoo.com
Ify = e@69S"* _ 1 < x < 1,showthat
(1x2) 2% ~~ ® _ ay =0
—x*) + -x—-a@y=0.
dx? dx y
y= et cos "1x
Differentiating w.r. t.x.
dy d(e* cos~1x )
dx . dx
-1
dy _ pacostx y d(acos~+x)
dx dx
dy — pacos tx ( a1 )
ax © *O\ Fe

Misc 19 teachoo.com
Using mathematical induction prove that < (x")=nx1
for all positive integers n.
Let P(): a (x")=nx"4
“dx
Forn = 1
Solving LHS
d(x+) dx
dx dx
=1
= RHS
Thus, P(n) is true form = 1

Ms. Remka of city school is teaching chain rule to her students with the help of a flow-chart The chain rule says that if h and g are functions and
f
(
x
) =
g(h(x
)), then
Based on the above information, answer any four of the following questions.
Let f(x) = sin x and g(x) = x3
Question 1
fog
(x) = _______.
(a) sin x3
(b) sin3 x
(c) sin 3x
(d) 3 sin x
Question 2
gof
(x) = _______.
(a) sin x3
(b) sin3 x
(c) sin 3x
(d) 3 sin x
Question 3
d
/
dx
(sin3 x) = _______.
(a) cos3 x
(b) 3 sinx cos x
(c) 3 sin2x cos x
(d) −cos3 x
Question 4
d
/
dx
(
sin x3) _______.
(a) cos (x)3
(b) −cos (x)3
(c) 3x2 sin (x)3
(d) 3x2 cos (x3)
Question 5
d
/
dx
(sin 2x) at x = π/2 is _______.
(a) 0
(b) 1
(c) 2
(d) −2

Rolle’s Theorem: Suppose following three condition hold for function y =
f
(x):
1. function is defined and continuous on closed interval [a, b];
2. exists finite derivative
f
‘(x) on interval (a, b);
3.
f
(a) =
f
(b).
then there exists point c(a < c < b) such that
f
‘(c) = 0.
Based on the above information, answer any four of the following questions.
Question 1
(i) Rolle’s theorem is not applicable for the function f(x) = tan x in [0, π] because _______.
(a) it is not continuous in [0, π]
(b) it is differentiable in (0, π)
(c) f(0) ≠
f
(π)
(d) f(0) =
f
(π)
Question 2
The value of c satisfying Rolle’s theorem for the function g(x) = sin x in [0, π] is _______.
(a) 0
(b) p
(c) π/2
(d) π/4
Question 3
The value of c satisfying Rolle’s theorem for the function h(x) = cos x in [0, 2π] is _______.
(a) 0
(b) π
(c) π/2
(d) 3π/2
Question 4
The value of c satisfying Rolle’s theorem for the function p(x) = sin x + cos x in [0, 𝜋] is _______.
(a) 0
(b) π
(c) π/4
(d) π/2
Question 5
Rolle’s theorem is not applicable for the function
f
(x) = |x| in [–2, 2] because _______.
(a)
f
(–2)¹ f(2)
(b)
f
(x) is not continuous in [–2, 2]
(c)
f
(x) is not differentiable in (–2, 2)
(d) None of these

The function f (x) = {■8(sin⁡𝑥/𝑥 " + cos x, if x " ≠" 0" @𝑘 ", if x " =" 0" )┤ is continuous at x = 0, then the value of k is
(A) 3 (B) 2
(C) 1 (D) 1.5
At 𝒙 = 0

The function f (x) = [x], where [x] denotes the greatest integer function, is continuous at
(A) 4 (B) −2
(C) 1 (D) 1.5
Given
𝑓(𝑥) = [𝑥]

The number of points at which the function f (x) = 1/(𝑥−[𝑥] ) is not continuous is
(A) 1 (B) 2
(C) 3 (D) none of these
Given f(x) = 1/(𝑥 − [𝑥] )

The function given by f (x) = tan x is discontinuous on the set
(A) {𝑛𝜋:𝑛 𝜖 𝒛} (B) {2𝑛𝜋:𝑛 𝜖 𝒛}
(C) {(2n + 1) 𝜋/2 : 𝑛 𝜖 𝒛} (D) {𝑛𝜋/2 " : " 𝑛 𝜖 𝒛}
𝑓(𝑥) = tan 𝑥
𝒇(𝒙) = 𝐬𝐢𝐧⁡𝒙/𝐜𝐨𝐬⁡𝒙

Let f (x) = |cos x|. Then,
(A) f is everywhere differentiable.
(B) f is everywhere continuous but not differentiable at n = n𝜋, n ∈ Z
(C) f is everywhere continuous but not differentiable at x = (2n + 1)𝜋/2, n ∈ Z
(D) None of these
f(𝑥) = |cos 𝑥|

The function f (x) = |x| + |x – 1| is
(A) continuous at x = 0 as well as at x = 1.
(B) continuous at x = 1 but not at x = 0.
(C) discontinuous at x = 0 as well as at x = 1.
(D) continuous at x = 0 but not at x = 1.
Given 𝑓(𝑥)= |𝑥|+|𝑥−1|

The value of k which makes the function defined by f (x) = {■8(𝑠𝑖𝑛 1/𝑥," if " 𝑥≠"0 " @𝑘 ", if x " ="0" )┤ , continuous at x = 0 is
8 (B) 1
(C) −1 (D) None of these

The set of points where the functions f given by f (x) = |x – 3| cos x is differentiable is
(A) R (B) R − {3}
(C) (0, ∞) (D) None of these
f(x) = |𝑥−3| cos⁡𝑥
= {█((𝑥−3) cos⁡𝑥, 𝑥−3≥0@−(𝑥−3) cos⁡𝑥, 𝑥−3<0)┤
= {█((𝑥−3) cos⁡𝑥, 𝑥≥3@−(𝑥−3) cos⁡𝑥, 𝑥<3)┤

Differential coefficient of sec (〖𝑡𝑎𝑛〗^(−1)x) w.r.t. x is
(A) 𝑥/√(1 + 𝑥^2 ) (B) 𝑥/(1 + 𝑥^2 )
(C) x √(1+𝑥^2 ) (D) 1/√(1 + 𝑥^2 )
Let y = sec (〖𝑡𝑎𝑛〗^(−1) 𝑥)
Differential coefficient sec (〖𝑡𝑎𝑛〗^(−1)x) means 𝒅𝒚/𝒅𝒙

If u = 〖𝑠𝑖𝑛〗^(−1) (2𝑥/(1 + 𝑥^2 )) and v = 〖𝑡𝑎𝑛〗^(−1) (2𝑥/(1 − 𝑥^2 )), then 𝑑𝑢/𝑑𝑣 is
(A) 1/2 (B) 𝑥
(C) (1 − 𝑥^2)/(1 + 𝑥^2 ) (D) 1
𝒅𝒖/𝒅𝒗=(𝒅𝒖⁄𝒅𝒙)/(𝒅𝒗⁄𝒅𝒙)

If f (x) = 2x and g (x) = 𝑥^2/2+1 , then which of the following can be a discontinuous function
(A) 𝑓 (𝑥) + 𝑔 (𝑥) (B) f (x) – g (x)
(C) 𝑓 (𝑥) . 𝑔 (𝑥) (D) 𝑔(𝑥)/(𝑓(𝑥))
Given functions
𝑓(𝑥)=2𝑥 & 𝑔(𝑥)=𝑥^2/2+1
Checking each option one by one

The function f (x) = (4 − 𝑥^2)/(4𝑥 −𝑥^3 ) is
(A) discontinuous at only one point
(B) discontinuous at exactly two points
(C) discontinuous at exactly three points
(D) none of these
f(x) = (4 − 𝑥^(2 ) )/(4𝑥 − 𝑥^3 )
= (2^2 − 𝑥^(2 ) )/(𝑥(4 − 𝑥^2))
= ((𝟐 − 𝒙)(𝟐 + 𝒙))/(𝒙(𝟐 − 𝒙)(𝟐 + 𝒙))

The set of points where the function f given by f (x) = |2x−1| sin x is differentiable is
R (B) R − {1/2} (C) (0, ∞) (D) none of these
f(x) = |2𝑥−1| sin⁡𝑥
= {█((2𝑥−1) sin⁡𝑥, 2𝑥−1≥0@−(2𝑥−1) sin⁡𝑥, 2𝑥−1<0)┤
= {█((2𝑥−1) sin⁡𝑥, 𝑥≥1/2@−(2𝑥−1) sin⁡〖𝑥 ,〗 𝑥<1/2)┤

The function f (x) = cot x is discontinuous on the set
(A) {𝑥=𝑛𝜋:𝑛∈𝒁} (B) {𝑥=2𝑛𝜋:𝑛∈𝒁}
(C) {𝑥=(𝟐𝒏+𝟏) 𝝅/𝟐 ;𝑛∈𝒛} (D) {𝑥=𝒏𝝅/𝟐 ;𝑛∈𝒛}
Let 𝑓(𝑥) = c𝑜𝑡⁡𝑥
𝒇(𝒙) = 𝒄𝒐𝒔⁡𝒙/𝒔𝒊𝒏⁡𝒙

The function f (x) = 𝑒^(|𝑥|) is
(A) continuous everywhere but not differentiable at x = 0
(B) continuous and differentiable everywhere
(C) not continuous at x = 0
(D) none of these.
f(𝑥) = 𝑒^(|𝑥|)

If f (x) = x2 sin 1/𝑥, where x ≠ 0, then the value of the function f at x = 0, so that the function is continuous at x = 0, is
(A) 0 (B) – 1
(C) 1 (D) none of these
Given
f (x) = x2 sin 1/𝑥, when x ≠ 0

If f (x) = {█(𝑚𝑥+1, 𝑖𝑓 𝑥≤𝜋/2@sin⁡〖𝑥+𝑛, 𝑖𝑓 𝑥> 𝜋/2〗 )┤ , is continuous at x = 𝜋/2, then
(A) m = 1, n = 0 (B) m = 𝑛𝜋/2 + 1
(C) n = 𝑚𝜋/2 (D) none of these
Given that function is continuous at 𝑥=𝜋/2

Let f (x) = |sin x|. Then
(A) f is everywhere differentiable
(B) f is everywhere continuous but not differentiable at x = n𝜋, n∈ Z.
(C) f is everywhere continuous but not differentiable at x = (2n + 1) 𝜋/2, n∈ Z.
(d) None of these
f(𝑥) = |sin 𝑥|

If y = log ((1 − 𝑥^2)/(1 + 𝑥^2 )),then 𝑑𝑦/𝑑𝑥 is equal to
(A) 〖4𝑥〗^3/(1−𝑥^4 ) (B) (−4𝑥)/(1−𝑥^4 )
(C) 1/(4−𝑥^4 ) (D) (−4𝑥^3)/(1−𝑥^4 )
y=log((1 − 𝑥^2)/(1 +〖 𝑥〗^2 ))
𝐲=𝐥𝐨𝐠(𝟏−𝒙^𝟐 )−𝐥𝐨𝐠⁡〖(𝟏+𝒙^𝟐)〗

If y = √(sin⁡〖𝑥+𝑦〗 ), then 𝑑𝑦/𝑑𝑥 is equal to
(A) cos⁡𝑥/(2𝑦−1) (B) cos⁡𝑥/(1−2𝑦)
(C) sin⁡𝑥/(1−2𝑦) (D) (−4𝑥^3)/(2𝑦 −1)
𝑦=√(𝑠𝑖𝑛⁡〖𝑥+𝑦〗 )
Squaring both sides
𝑦^2=(√(sin⁡〖𝑥+𝑦〗 ))^2
𝒚^𝟐=𝒔𝒊𝒏⁡〖𝒙+𝒚〗
𝑦^2−𝑦=sin⁡𝑥

The derivative of cos–1 (2x2 – 1) w.r.t. cos–1x is
(A) 2 (B) (−1)/(2√(1−𝑥^2 ))
(C) 2/𝑥 (D) 1 − x2
Let 𝑦=〖𝑐𝑜𝑠〗^(−1)⁡𝑥
cos⁡〖𝑦=𝑥〗
〖𝒙=𝒄𝒐𝒔〗⁡𝒚

If x = t2, y = t3, then (𝑑^2 𝑦)/(𝑑𝑥^2 ) is
(A) 3/2 (B) 3/4𝑡
(C) 3/2𝑡 (D) 3/4
𝑥=𝑡^2 & 𝑦=𝑡^3

The value of c in Rolle’s Theorem for the function f (x) = 𝑒^𝑥 sin 𝑥,
𝑥∈ [0, 𝜋] is
(A) 𝜋/6 (B) 𝜋/4 (C) 𝜋/2 (D) 3𝜋/4
𝑓 (𝑥)= 𝑒^𝑥 sin⁡〖𝑥, 𝑥 ∈ [0,𝜋] 〗

The value of c in Mean value theorem for the function f (x) = x (x – 2), x ∈ [1, 2] is
(A) 3/2 (B) 2/3 (C) 1/2 (D) 3/2
𝑓(𝑥)=𝑥" (" 𝑥" – 2)"
𝑓(𝑥) = 𝑥^2 – 2𝑥 in interval [1, 2].
Checking conditions for
Mean value Theorem

The value of c in Rolle’s theorem for the function f (x) = x3 – 3x in the interval [0,√3] is
(A) 1 (B) −1
(C) 3/2 (D) 1/3
𝑓 (𝑥)= 𝑥3 −3𝑥 , 𝑥 ∈ [0,√3]

For the function f (x) = x + 1/𝑥 , x ∈ [1, 3], the value of c for mean value theorem is
1 (B) √3
(C) 2 (D) None of these
𝑓(𝑥)="x + " 1/𝑥 in interval [1, 3]
Checking conditions for
Mean value Theorem

Ex 5.8, 1 teachoo
Verify Rollie’s theorem for the function
f (x) = x? + 2x- 8x € [4,2].
Let’s check conditions of Rolle’s theorem
Conditions of Rolle’s theorem
1. f(x) is continuous at [a , b]
2. f(x) is derivable at (a,b)
Condition 1 3 f(@= f(b)
We need to check If all 3 conditions are satisfied
if f (x) is continuous at [- 4, 2] then there exist some cin (a,b)
such that f'(c) =0
Since f(x) = x? + 2x - 8 isa polynomial
& Every polynomial function is continuous for all x € R
«. f ()is continuous at x € [- 4, 2]

Ex 5.8, 2 teachoo.com
Examine if Rolle’s theorem is applicable to the functions. Can you say
some thing about the converse of Rolle’s theorem from this function 3
() f @) = [x] forx € [5,9]
Greatest Integer less than equal to x
Value of x Value of Going by Same concept
Greatest Value of x Value of
Integer Greatest Integer
3 3 c c
3.1 3 ct c
3.9999 3 -
c c-1
2.9999 2
f @) = [x] is not continuous & differentiable
= Condition of Rolle’s Theorem is not satisfied.
Therefore, Rolle’s Theorem is not applicable .

Examine if Rolle’s theorem is applicable to the functions. Can you say some thing about the converse of Rolle’s theorem from this function? (𝑖𝑖) 𝑓 (𝑥) = [𝑥] 𝑓𝑜𝑟 𝑥 ∈ [−2, 2]Greatest Integer less than equal to 𝑥

Examine if Rolle’s theorem is applicable to the functions. Can you say some thing about the converse of Rolle’s theorem from this function? (𝑖𝑖𝑖) 𝑓 (𝑥) = 𝑥2 – 1 𝑓𝑜𝑟 𝑥 ∈ [1, 2]𝑓 (𝑥) = 𝑥2 – 1 𝑓𝑜𝑟 𝑥 ∈ [1 , 2]

Ex 5.8, 3 teachoo.com
lf ff: [- 5,5] — Ris adifferentiable function and if f ’(x) does
not vanish anywhere, then prove that f (-5) # f (5).
f: [- 5,5] > Risa differentiable
=> We know that every differentiable function is continuous.
Therefore f is continuous & differentiable both on (-5, 5}
By Mean Value Theorem
There exist some c in (5, -5}

b =
Such that f’(c) = fO)- ©)

Ex 5.8, 4 teachoo.com
Verify Mean Value Theorem, if f (x) = x2- 4x - 3 inthe
interval [a,b], wherea = land b= 4
f (*) = x?-4x-3
x € [a,b] wherea=1&b=4 Conditions of Mean value theorem
1. f(x) is continuous at (a,b)
2. f(x) is derivable at (a,b)
Mean Value Theorem tf both conditions satisfied, then
satisfied if there exist some cin (a,b)
such that f'(c) = AO WTO io)
Condition 1
f @&) is continuous
f(x) = x?- 4x- 3
f (x) is a polynomial & Every polynomial function is continuous
> f(x) is continuous at x € [1,4]

Ex 5.8, 5 teachoo.com
Verify Mean Value Theorem, if f (x) = x°- 5x?- 3x in the interval
[a, b], where a = 1 and 6 = 3. Find allc € (1,3) for which f ’(c) = 0.
f (x) = x*8- 5x?- 3xin [a, 5],
where a=1andb=3 Conditions of Mean value
theorem
Condition 1 f (x) is continuous at (a, b)
2. f(x) is derivable at (a, b)
f (x) = x8- 5x?- 3x
If both conditions satisfied, then
i | ial &
F(%) is a polynomial & every there exist some cin (a,b)
polynomial function is continuous
such that f'(c) = fe) ~ f@)
- f(x) is continuous at x € [1,3] b-a

teachoo.com
Ex 5.8, 6
Examine the applicability of Mean Value Theorem in the function
Ci) f (x) = [x] for x € [-2,2]
Greatest Integer less than equal to x
Value of x Value of Going by Same concept
Greatest Value of x Value of
Integer Greatest Integer
3 3 c c
31 3 ct c
3.9999 3 -
c c-1
2.9999 2
f @) = [x] is not continuous & differentiable
=> Condition of Mean Value Theorem is not satisfied.
Therefore, Mean Value Theorem is not applicable .

Chapter 5 Class 12 - Continuity & Differentiability
- MCQ Worksheet 1
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Chapter 5 Class 12 - Continuity & Differentiability
- Assertion and Reasoning
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Chapter 5 Class 12 - Continuity & Differentiability
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