# Example 45

Last updated at May 29, 2018 by Teachoo

Last updated at May 29, 2018 by Teachoo

Transcript

Example 45 Differentiate the following 𝑤.𝑟.𝑡. 𝑥. (i) cos−1 (sin𝑥) Let 𝑓(𝑥) = cos−1 (sin𝑥) 𝑓(𝑥) = cos−1 cos 𝜋2 −𝑥 𝒇(𝒙) = 𝝅𝟐 −𝒙 Differentiating 𝑤.𝑟.𝑡.𝑥 𝑓’(𝑥) = 𝑑 𝜋2𝑑𝑥 − 𝑑(𝑥)𝑑𝑥 𝑓’(𝑥) = 0 − 1 𝒇’(𝒙) = − 1 Example 45 Differentiate the following w.r.t. x. (ii) tan −1 sin𝑥 1 + cos𝑥 Let 𝑓(𝑥) = tan −1 sin𝑥 1 + cos𝑥 𝑓(𝑥) = tan −1 2 sin 𝑥2 cos 𝑥2 1+ (2 cos2 𝑥2 − 1) = tan −1 2 sin 𝑥2 cos 𝑥2 2 cos2 𝑥2 = tan −1 2 sin 𝑥2 2 cos 𝑥2 = tan −1 sin 𝑥2 cos 𝑥2 = tan −1 tan 𝑥2 = 𝑥2 𝒇(𝒙) = 𝒙𝟐 Differentiating 𝑤.𝑟.𝑡.𝑥 𝑓’(𝑥) = 12 𝑑(𝑥)𝑑𝑥 𝒇’(𝒙) = 𝟏𝟐 Example 45 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 + 2𝑥2 Let 𝟐𝒙 = tan θ 𝑓(𝑥) = sin−1 tan𝜃 . 2 1 + tan2𝜃 = sin−1 2 tan𝜃 1 + tan2𝜃 = sin−1 (sin 2𝜃) = 2𝜃 ∴ 𝑓(𝑥) = 2𝜃 As 2𝑥= tan𝜃 tan−1 2𝑥=𝜃 ∴ 𝒇(𝒙) = 𝟐 𝒕𝒂𝒏−𝟏 𝟐𝒙 Differentiating 𝑤.𝑟.𝑡.𝑥 𝑓’(𝑥) = 2 𝑑 tan−1 2𝑥 𝑑𝑥 𝑓’(𝑥) = 2 . 11 + 2𝑥2 . 𝒅 𝟐𝒙 𝒅𝒙 𝑓’(𝑥) = 2 1 + 2𝑥2 . 𝟐𝒙 . 𝒍𝒐𝒈𝟐 𝑓’(𝑥) = 2. 2𝑥. log21 + 2𝑥2 𝑓’(𝑥) = 2𝑥 + 1. log21 + 2𝑥2 𝑓’(𝑥) = 2𝑥 + 1. log21 + 22𝑥 𝒇’(𝒙) = 𝟐𝒙 + 𝟏. 𝒍𝒐𝒈𝟐𝟏 + 𝟒𝒙

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Example 32 Important

Example 33 Important

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Example 41 Important

Ex 5.7, 14 Important

Example 42 Important

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Example 44 Important

Example 45 Important You are here

Example 47 Important

Misc 6 Important

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Class 12

Important Question for exams Class 12

- Chapter 1 Class 12 Relation and Functions
- Chapter 2 Class 12 Inverse Trigonometric Functions
- Chapter 3 Class 12 Matrices
- Chapter 4 Class 12 Determinants
- Chapter 5 Class 12 Continuity and Differentiability
- Chapter 6 Class 12 Application of Derivatives
- Chapter 7 Class 12 Integrals
- Chapter 8 Class 12 Application of Integrals
- Chapter 9 Class 12 Differential Equations
- Chapter 10 Class 12 Vector Algebra
- Chapter 11 Class 12 Three Dimensional Geometry
- Chapter 12 Class 12 Linear Programming
- Chapter 13 Class 12 Probability

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.