Misc 28 - Chapter 12 Class 11 Limits and Derivatives

Transcript

Misc 28 Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 𝑥/(1 + 𝑡𝑎𝑛⁡𝑥 ) Let f (x) = 𝑥/(1 + 𝑡𝑎𝑛⁡𝑥 ) Let u = x & v = 1 + tan x So, f(x) = 𝑢/𝑣 ∴ f’(x) = (𝑢/𝑣)^′ Using quotient rule f’(x) = (𝑢^′ 𝑣 −〖 𝑣〗^′ 𝑢)/𝑣^2 Finding u’ & v’ u = x u’ = 1 & v = 1 + tan x v’ = (1 + tan x)’ = 0 + sec2 x = sec2 x Now, f’(x) = (𝑢/𝑣)^′ = (𝑢^′ 𝑣 −〖 𝑣〗^′ 𝑢)/𝑣^2 = (1(1 +〖 tan〗⁡〖𝑥)〗 − 𝑠𝑒𝑐2 𝑥 (𝑥))/〖(1 +〖 tan〗⁡〖𝑥)〗〗^2 = (𝟏 +〖 𝒕𝒂𝒏〗⁡〖𝒙 − 𝒙 𝒔𝒆𝒄𝟐 𝒙〗)/〖(𝟏 +〖 𝒕𝒂𝒏〗⁡〖𝒙)〗〗^𝟐 = (𝑢^′ 𝑣 −〖 𝑣〗^′ 𝑢)/𝑣^2 = (1(1 +〖 tan〗⁡〖𝑥)〗 − 𝑠𝑒𝑐2 𝑥 (𝑥))/〖(1 +〖 tan〗⁡〖𝑥)〗〗^2 = (𝟏 +〖 𝒕𝒂𝒏〗⁡〖𝒙 − 𝒙 𝒔𝒆𝒄𝟐 𝒙〗)/〖(𝟏 +〖 𝒕𝒂𝒏〗⁡〖𝒙)〗〗^𝟐