# Ex 5.7, 10 - Chapter 5 Class 12 Continuity and Differentiability

Last updated at Jan. 3, 2020 by Teachoo

Last updated at Jan. 3, 2020 by Teachoo

Transcript

Ex 5.7, 10 Find the second order derivatives of the function 〖 sin〗〖 (log〖𝑥)〗 〗 Let y = 〖 sin〗〖 (log〖𝑥)〗 〗 Differentiating 𝑤.𝑟.𝑡.𝑥 . 𝑑𝑦/𝑑𝑥 = (𝑑(〖 sin〗〖 (log〖𝑥)〗 〗))/𝑑𝑥 𝑑𝑦/𝑑𝑥 = cos(log𝑥) . (𝑑(log〖𝑥)〗)/𝑑𝑥 𝑑𝑦/𝑑𝑥 = cos(log𝑥) . 1/𝑥 𝑑𝑦/𝑑𝑥 = (cos(log𝑥))/𝑥 Again Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑/𝑑𝑥 (𝑑𝑦/𝑑𝑥) = 𝑑/𝑑𝑥 ((cos(log𝑥))/𝑥) (𝑑^2 𝑦)/(𝑑𝑥^2 ) = 𝑑/𝑑𝑥 ((cos(log𝑥))/𝑥) (𝑑^2 𝑦)/(𝑑𝑥^2 ) = ((𝑑(cos(log𝑥)))/𝑑𝑥 . 𝑥 − (𝑑 (𝑥))/𝑑𝑥 . cos(log𝑥))/𝑥^2 (𝑑^2 𝑦)/(𝑑𝑥^2 ) = (−〖sin 〗(log𝑥 ) . 𝑑(log𝑥 )/𝑑𝑥 . 𝑥 − 1 . cos(log𝑥))/𝑥^2 Using Quotient Rule As, (𝑢/𝑣)^′= (𝑢’𝑣 − 𝑣’𝑢)/𝑣^2 where v = cos (log x) & v = x (𝑑^2 𝑦)/(𝑑𝑥^2 ) = (−〖sin 〗(log𝑥 ) . 1/𝑥 . 𝑥 − cos (log𝑥))/𝑥^2 (𝑑^2 𝑦)/(𝑑𝑥^2 ) = (−〖sin 〗(log𝑥 ) − cos(log𝑥))/𝑥^2 (𝒅^𝟐 𝒚)/(𝒅𝒙^𝟐 ) = (− [〖𝒔𝒊𝒏 〗(𝒍𝒐𝒈𝒙 ) + 𝒄𝒐𝒔(𝒍𝒐𝒈𝒙)])/𝒙^𝟐

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
- Finding second order derivatives - Normal form
- Finding second order derivatives- Implicit form
- Proofs
- Verify Rolles theorem
- Verify Mean Value Theorem

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.