# Example 44

Last updated at March 11, 2017 by Teachoo

Last updated at March 11, 2017 by Teachoo

Transcript

Example 44 Differentiate w.r.t. x, the following function: (i) 3𝑥+2 + 1 2 𝑥2+ 4 Let y = 3𝑥+2 + 1 2 𝑥2+ 4 Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑𝑦𝑑𝑥 = 𝑑 3𝑥+2 + 1 2 𝑥2+ 4 𝑑𝑥 𝑑𝑦𝑑𝑥 = 𝑑 3𝑥+2𝑑𝑥 + 𝑑 1 2 𝑥2+ 4 𝑑𝑥 𝑑𝑦𝑑𝑥 = 𝑑 3𝑥 + 2𝑑𝑥 + 𝑑 2 𝑥2 + 4 −12𝑑𝑥 Calculating 𝑑 3𝑥 + 2𝑑𝑥 & 𝑑 2 𝑥2 + 4 −12𝑑𝑥 separately Calculating 𝐝 𝟑𝐱 + 𝟐𝒅𝒙 𝑑 3𝑥 + 2𝑑𝑥 = 12 3𝑥 + 2 × 𝑑 3𝑥 + 2𝑑𝑥 = 12 3𝑥 + 2 × 3+0 = 32 3𝑥 + 2 Calculating 𝒅 𝟐 𝒙𝟐 + 𝟒 −𝟏𝟐𝒅𝒙 𝑑 2𝑥 + 4 −12𝑑𝑥 = −12 (2𝑥+4) −1 2 −1 . 𝑑 2 𝑥2+ 4𝑑𝑥 = −12 2 𝑥2+ 4 −3 2 . 𝑑 2 𝑥2+ 4𝑑𝑥 = −12 2 𝑥2+ 4 −3 2 . 𝑑 2 𝑥2𝑑𝑥 + 𝑑 4𝑑𝑥 = −12 2 𝑥2+ 4 −3 2 . 4𝑥+0 = −4𝑥2 2 𝑥2+ 4 −3 2 = −2𝑥 2 𝑥2+ 4 32 Hence, 𝑑𝑦𝑑𝑥 = 𝑑 3𝑥+2𝑑𝑥 + 𝑑 1 2 𝑥2+ 4 𝑑𝑥 𝒅𝒚𝒅𝒙 = 𝟑𝟐 𝟑𝒙 + 𝟐 − 𝟐𝒙 𝟐 𝒙𝟐+ 𝟒 −𝟑 𝟐 Example 44 Differentiate w.r.t. x, the following function: (ii) 𝑒 sec2𝑥 + 3 cos–1 𝑥 Let y = 𝑒 sec2𝑥 + 3 cos–1 𝑥 Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑𝑦𝑑𝑥 = 𝑑 𝑒 sec2𝑥 + 3 cos–1 𝑥 𝑑𝑥 𝑑𝑦𝑑𝑥 = 𝑑 𝑒 sec2𝑥 𝑑𝑥 + 𝒅 𝟑 𝒄𝒐𝒔–𝟏 𝒙𝒅𝒙 𝑑𝑦𝑑𝑥 = 𝑒 sec2𝑥 𝑑 sec2𝑥𝑑𝑥 + 3. −𝟏 𝟏 − 𝒙𝟐 𝑑𝑦𝑑𝑥 = 𝑒 sec2𝑥 . 2 sec 𝑥 . 𝒅 𝒔𝒆𝒄 𝒙𝒅𝒙 − 3 1 − 𝑥2 𝑑𝑦𝑑𝑥 = 𝑒 sec2𝑥 . 2 sec 𝑥 . 𝒔𝒆𝒄𝒙 . 𝒕𝒂𝒏𝒙 − 3 1 − 𝑥2 𝒅𝒚𝒅𝒙 = 𝟐 𝒆 𝒔𝒆𝒄𝟐𝒙 . 𝒔𝒆𝒄𝟐𝒙 . 𝒕𝒂𝒏𝒙 − 𝟑 𝟏 − 𝒙𝟐 Example 44 Differentiate w.r.t. x, the following function: (iii) log7 (log x) y = log7 (log x) But we do not solve base 7. So, changing base y = log7 ( log x) y = 𝐥𝐨𝐠( 𝐥𝐨𝐠𝒙) 𝐥𝐨𝐠𝟕 Now, differentiating 𝑑𝑦𝑑𝑥= 𝑑 ( log( log7)log 7𝑑𝑥 𝑑𝑦𝑑𝑥= 𝟏 𝐥𝐨𝐠𝟕 𝑑 log( log𝑥) 𝑑𝑥 𝑑𝑦𝑑𝑥= 1 log7. 1 log𝑥 𝑑( log𝑥)𝑑𝑥 𝑑𝑦𝑑𝑥= 1 log7. 1 log𝑥. 1𝑥 𝒅𝒚𝒅𝒙= 𝟏 𝐱 𝒍𝒐𝒈𝟕 𝐥𝐨𝐠𝒙

Ex 5.1, 9
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Example 32 Important

Example 33 Important

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Ex 5.5, 16 Important

Ex 5.6, 7 Important

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Example 41 Important

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Example 42 Important

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Example 44 Important You are here

Example 45 Important

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

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