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Ex 5.7, 8 - Chapter 5 Class 12 Continuity and Differentiability
Last updated at April 16, 2024 by Teachoo
Finding second order derivatives - Normal form
Chapter 5 Class 12 Continuity and Differentiability
Concept wise
- Checking continuity at a given point
- Checking continuity at any point
- Checking continuity using LHL and RHL
- Algebra of continous functions
- Continuity of composite functions
- Checking if funciton is differentiable
- Finding derivative of a function by chain rule
- Finding derivative of Implicit functions
- Finding derivative of Inverse trigonometric functions
- Finding derivative of Exponential & logarithm functions
- Logarithmic Differentiation - Type 1
- Logarithmic Differentiation - Type 2
- Derivatives in parametric form
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Finding second order derivatives - Normal form
- Finding second order derivatives- Implicit form
- Proofs
- Verify Rolles theorem
- Verify Mean Value Theorem
Transcript
Ex 5.7, 8 Find the second order derivatives of the function 〖𝑡𝑎𝑛〗^(−1) 𝑥 Let y = 〖𝑡𝑎𝑛〗^(−1) 𝑥 Differentiating 𝑤.𝑟.𝑡.𝑥 . 𝑑𝑦/𝑑𝑥 = (𝑑(〖𝑡𝑎𝑛〗^(−1) 𝑥))/𝑑𝑥 𝑑𝑦/𝑑𝑥 = 1/(1 + 𝑥^2 ) Again Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑/𝑑𝑥 (𝑑𝑦/𝑑𝑥) = 𝑑/𝑑𝑥 (1/(1 + 𝑥^2 )) (𝑑^2 𝑦)/(𝑑𝑥^2 ) = 𝑑/𝑑𝑥 (1/(1 + 𝑥^2 )) Using Quotient Rule As, (((𝑢)′)/𝑣) = (𝑢’𝑣 − 𝑣’𝑢)/𝑣^2 where u = 1 & v = 1 + x2 (𝑑^2 𝑦)/(𝑑𝑥^2 ) = ((𝑑(1))/𝑑𝑥 (1+𝑥^2 ) − (𝑑 (1 +𝑥^2 ))/𝑑𝑥 . 1 )/(1+𝑥^2 )^2 (𝑑^2 𝑦)/(𝑑𝑥^2 ) = (0 . (1+𝑥^2 ) − ((𝑑(1))/𝑑𝑥 + (𝑑〖(𝑥〗^2))/𝑑𝑥). 1 )/(1+𝑥^2 )^2 (𝑑^2 𝑦)/(𝑑𝑥^2 ) = (0 − ( 0 + 2𝑥 ) 1)/(1+𝑥^2 )^2 (𝒅^𝟐 𝒚)/(𝒅𝒙^𝟐 ) = (−𝟐𝒙)/(𝟏+𝒙^𝟐 )^𝟐
