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JEE Main
Exam
JEE Main
Subject
Mathematics
Chapter
Differentiation

JEE Main · Mathematics · Differentiation

If & ( cos x ) ^ y = ( y ) ^ x ,...

If & {( cos\text{ }x )}^{y} = {( \sin y )}^{x} , \text{ }find\text{ }\dfrac{dy}{dx} .

Options

  1. A

    & log⁡cos⁡y+ytanxlog⁡cos⁡x−xcoty\dfrac{\log \cos y + ytanx}{\log \cos x − xcoty}

  2. B

    & log⁡sin⁡y+ytanxlog⁡cos⁡x−xcoty\dfrac{\log \sin y + ytanx}{\log \cos x − xcoty}

  3. C

    & log⁡sin⁡y−ytanxlog⁡cos⁡x+xcoty\dfrac{\log \sin y − ytanx}{\log \cos x + xcoty}

  4. D

    & log⁡sin⁡y+ytanxlog⁡sin⁡x−xcoty\dfrac{\log \sin y + ytanx}{\log \sin x − xcoty}

Correct answer

B.

& log⁡sin⁡y+ytanxlog⁡cos⁡x−xcoty\dfrac{\log \sin y + ytanx}{\log \cos x − xcoty}

Step-by-step solution

We have,

(cos⁡x)y=(sin⁡y)x{( \cos x )}^{y} = {( \sin y )}^{x}

Taking logarithms on both sides, we get

~y\text{ }log\text{ }\cos x = xlogsiny

Differentiate with respect to x we get

~y\text{ }log\text{ }\cos x = xlogsiny

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