Check Full Chapter Explained - Continuity and Differentiability - Continuity and Differentiability Class 12

Last updated at April 8, 2019 by Teachoo

Check Full Chapter Explained - Continuity and Differentiability - Continuity and Differentiability Class 12

Transcript

Find the derivative of f given by f (x) = cosec–1 𝑥 assuming it exists. Let 𝑦 = cosec^(–1) 𝑥 cosec𝑦= 𝑥 𝑥 =cosec𝑦 Differentiating both sides 𝑤.𝑟.𝑡.𝑥 (𝑑(𝑥))/𝑑𝑥 = (𝑑 (cosec𝑦 ))/𝑑𝑥 1 = (𝑑 (cosec𝑦 ))/𝑑𝑥 We need 𝑑𝑦 in denominator, so multiplying & Dividing by 𝑑𝑦. 1 = (𝑑 (cosec𝑦 ))/𝑑𝑥 × 𝑑𝑦/𝑑𝑦 1 = cosec𝑦 .cot𝑦 . 𝑑𝑦/𝑑𝑥 〖−cosec〗𝑦 .cot𝑦 𝑑𝑦/𝑑𝑥 = 1 𝑑𝑦/𝑑𝑥 = 1/(〖−cosec〗𝑦 .𝒄𝒐𝒕𝒚 ) 𝑑𝑦/𝑑𝑥 = 1/(〖−cosec〗𝑦 . √(〖𝐜𝐨𝒔𝒆𝒄〗^𝟐𝒚−𝟏)) 𝑊𝑒 𝑘𝑛𝑜𝑤 𝑡ℎ𝑎𝑡 cot^2〖𝜃=cosec^2〖𝜃−1〗 〗 cot𝜃= √(cosec^2𝜃−1) Putting value of 𝑐𝑜𝑠𝑒𝑐𝑦 = 𝑥 𝑑𝑦/𝑑𝑥 = 1/(− 𝑥√(𝑥^2 − 1 )) As 𝑦 = cosec^(−1) 𝑥 So, co𝑠𝑒𝑐𝑦 = 𝑥 𝑑𝑦/𝑑𝑥 = (−1)/(𝑥 √(𝑥^2 − 1 ) ) Hence 𝒅(〖𝒄𝒐𝒔𝒆𝒄〗^(–𝟏) 𝒙)/𝒅𝒙 = (−𝟏)/(𝒙 √(𝒙^𝟐 − 𝟏 ) )

Finding derivative of Inverse trigonometric functions

Derivative of cos-1 x (Cos inverse x)

Example 26 Important

Example 27

Derivative of cot-1 x (cot inverse x)

Derivative of sec-1 x (Sec inverse x)

Derivative of cosec-1 x (Cosec inverse x) You are here

Ex 5.3, 14

Ex 5.3, 9 Important

Ex 5.3, 13 Important

Ex 5.3, 12 Important

Ex 5.3, 11 Important

Ex 5.3, 10 Important

Ex 5.3, 15 Important

Misc 5 Important

Misc 4

Misc 13 Important

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.