# Example 33

Last updated at March 11, 2017 by Teachoo

Last updated at March 11, 2017 by Teachoo

Transcript

Example 33 Find 𝑑𝑦𝑑𝑥 , if 𝑦𝑥 + 𝑥𝑦 + 𝑥𝑥=𝑎𝑏 . Let u = 𝑦𝑥, v = 𝑥𝑦 & w = 𝑥𝑥 Now, 𝑢 + 𝑣 + 𝑤 = 𝑎𝑏 Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑 (𝑢 + 𝑣 + 𝑤)𝑑𝑥 = 𝑑( 𝑎𝑏)𝑑𝑥 𝑑(𝑢)𝑑𝑥 + 𝑑(𝑣)𝑑𝑥 + 𝑑(𝑤)𝑑𝑥 = 0 We will calculate derivative of u, v & w separately . Finding Derivative of 𝒖 . 𝑢 = 𝑦𝑥 Taking log both sides log𝑢= log ( 𝑦𝑥) log𝑢= 𝑥 . log𝑦 Differentiating both sides 𝑤.𝑟.𝑡.𝑥 𝑑( log𝑢)𝑑𝑥 = 𝑑(𝑥 . log𝑦)𝑑𝑥 𝑑( log𝑢)𝑑𝑥 𝑑𝑢𝑑𝑢 = 𝑑 𝑥. log𝑦𝑑𝑥 1𝑢 . 𝑑𝑢𝑑𝑥 = 𝑑 𝑥 . log𝑦𝑑𝑥 1𝑢 . 𝑑𝑢𝑑𝑥 = 𝑑 𝑥𝑑𝑥 . log𝑦 + 𝑑( log𝑦)𝑑 . 𝑥 1𝑢 . 𝑑𝑢𝑑𝑥 = 𝑑 𝑥𝑑𝑥 . log𝑦 + 𝑑( log𝑦)𝑑 . 𝑥 1𝑢 . 𝑑𝑢𝑑𝑥 = 1 . log𝑦 + 𝑥. 𝑑 log𝑦𝑑𝑥 . 𝑑𝑦𝑑𝑦 1𝑢 . 𝑑𝑢𝑑𝑥 = log𝑦 + 𝑥. 𝑑 log𝑦𝑑𝑥 . 𝑑𝑦𝑑𝑥 1𝑢 . 𝑑𝑢𝑑𝑥 = log𝑦 + 𝑥. 1𝑦 . 𝑑𝑦𝑑𝑥 1𝑢 . 𝑑𝑢𝑑𝑥 = log𝑦 + 𝑥𝑦 . 𝑑𝑦𝑑𝑥 𝑑𝑢𝑑𝑥 = 𝑢 log𝑦 + 𝑥𝑦 𝑑𝑦𝑑𝑥 𝑑𝑢𝑑𝑥 = 𝑦𝑥 log𝑦 + 𝑥𝑦 𝑑𝑦𝑑𝑥 Finding derivative of v v = xy Taking log both sides log𝑣= log ( 𝑥𝑦) log𝑣= 𝑦. log𝑥 Differentiating both sides 𝑤.𝑟.𝑡.𝑥 𝑑( log𝑣)𝑑𝑥 = 𝑑(𝑦 . log𝑥)𝑑𝑥 𝑑( log𝑣)𝑑𝑥 𝑑𝑣𝑑𝑥 = 𝑑 𝑦 log𝑥𝑑𝑥 1𝑣 𝑑𝑣𝑑𝑥 = 𝑑 𝑦 log𝑥𝑑𝑥 1𝑣 𝑑𝑣𝑑𝑥 = 𝑑 𝑦𝑑𝑥 . log𝑥 + 𝑑 ( log𝑥)𝑑𝑥 . 𝑦 1𝑣 𝑑𝑣𝑑𝑥 = 𝑑 𝑦𝑑𝑥 . log𝑥 + 𝑑 ( log𝑥)𝑑𝑥 . 𝑦 1𝑣 𝑑𝑣𝑑𝑥 = 𝑑𝑦𝑑𝑥 . log𝑥 + 1𝑥 . 𝑦 1𝑣 𝑑𝑣𝑑𝑥 = 𝑑𝑦𝑑𝑥 log𝑥 + 𝑦𝑥 𝑑𝑣𝑑𝑥 = v log 𝑑𝑦𝑑𝑥 𝑥+ 𝑦𝑥 Putting values of 𝑣 = 𝑥𝑦 𝑑𝑣𝑑𝑥 = 𝑥𝑦 𝑑𝑦𝑑𝑥 log𝑥+ 𝑦𝑥 Calculating derivative of 𝒘 𝑤 = 𝑥𝑥 Taking log both sides log𝑤= log ( 𝑥𝑥) log𝑤= 𝑥. log𝑥 log𝑤= 𝑥. log𝑥 Differentiating both sides 𝑤.𝑟.𝑡.𝑥 𝑑( log𝑤)𝑑𝑥 = 𝑑(𝑥 . log𝑥)𝑑𝑥 𝑑( log𝑤)𝑑𝑥 𝑑𝑤𝑑𝑤 = 𝑑 𝑥 log𝑥𝑑𝑥 𝑑( log𝑤)𝑑𝑤 . 𝑑𝑤𝑑𝑥 = 𝑑 𝑥 log𝑥𝑑𝑥 1𝑤 . 𝑑𝑤𝑑𝑥 = 𝑑 𝑥 log𝑥𝑑𝑥 1𝑤 𝑑𝑤𝑑𝑥 = 𝑑 𝑥𝑑𝑥 . log𝑥 + 𝑑 ( log𝑥)𝑑𝑥 . 𝑥 1𝑤 𝑑𝑤𝑑𝑥 = 1 . log𝑥 + 1𝑥 . 𝑥 1𝑤 𝑑𝑤𝑑𝑥 = ( log𝑥+1) 𝑑𝑤𝑑𝑥 = 𝑤( log𝑥+1) 𝑑𝑤𝑑𝑥 = 𝑥𝑥 log𝑥+1 From (1) 𝑑𝑢𝑑𝑥 + 𝑑𝑣𝑑𝑥 + 𝑑𝑤𝑑𝑥 = 0 Putting values from (2), (3) & (4) 𝑦𝑥 log𝑦+ 𝑦𝑥−1. 𝑥 𝑑𝑦𝑑𝑥 + 𝑥𝑦 log𝑥. 𝑑𝑦𝑑𝑥+ 𝑥𝑦. 𝑦𝑥 + 𝑥𝑥( log𝑥+1)=0 𝑦𝑥 log𝑦+ 𝑥𝑦. 𝑦𝑥+ 𝑥𝑥 ( log𝑥+1) + 𝑦𝑥−1 .𝑥 𝑑𝑦𝑑𝑥+ 𝑥𝑦 log𝑥 𝑑𝑦𝑑𝑥 = 0 𝑦𝑥−1 .𝑥 𝑑𝑦𝑑𝑥+ 𝑥𝑦 log𝑦 𝑑𝑦𝑑𝑥 = − 𝑦𝑥 log𝑦+ 𝑥𝑦. 𝑦𝑥+ 𝑥𝑥 ( log𝑥+1) 𝑦𝑥−1 .𝑥 + 𝑥𝑦 log𝑥 𝑑𝑦𝑑𝑥 = − 𝑦𝑥 log𝑦+ 𝑥𝑦. 𝑦𝑥+ 𝑥𝑥 ( log𝑥+1) 𝒅𝒚𝒅𝒙 = − 𝒚𝒙 𝒍𝒐𝒈𝒚 + 𝒙𝒚. 𝒚𝒙 + 𝒙𝒙 (𝟏 + 𝒍𝒐𝒈𝒙)( 𝒙𝒚𝒙−𝟏 + 𝒙𝒚 𝒍𝒐𝒈𝒙 )

Ex 5.1, 9
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Example 32 Important

Example 33 Important You are here

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Example 41 Important

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Example 42 Important

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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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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 7 years. He provides courses for Mathematics and Science from Class 6 to 12. You can learn personally from here https://www.teachoo.com/premium/maths-and-science-classes/.