CBSE (Science) Class 12CBSE
Share
Notifications

View all notifications
Books Shortlist
Your shortlist is empty

Solution for Differentiate ( Sin X ) Cos X ? - CBSE (Science) Class 12 - Mathematics

Login
Create free account


      Forgot password?

Question

Differentiate \[\left( \sin x \right)^{\cos x}\] ?

Solution

\[\text{ Let y} = \left( \sin x \right)^{\cos x }. . . \left( i \right)\]
\[\text{ Taking \log on both sides}, \]
\[\log y = \log \left( \sin x \right)^{\cos x }\]
\[ \Rightarrow \log y = \cos x \log \sin x \]
\[\text{ Differentiating with respect to x }, \]
\[\frac{1}{y}\frac{dy}{dx} = \cos x\frac{d}{dx}\left( \log \sin x \right) + \log \sin x\frac{d}{dx}\left( \cos x \right) \]
\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = \cos x\frac{1}{\sin x}\frac{d}{dx}\left( \sin x \right) + \log \sin x\left( - \sin x \right)\]
\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = \frac{\cos x}{\sin x}\left( \cos x \right) - \sin x \log \sin x\]
\[ \Rightarrow \frac{dy}{dx} = y\left[ \cos x \cot x - \sin x \log \sin x \right]\]
\[ \Rightarrow \frac{dy}{dx} = \left( \sin x \right)^{\cos x} \left[ \cos x \cot x - \sin x \log \sin x \right] \left[ \text{using equation }\left( i \right) \right]\]

  Is there an error in this question or solution?

Video TutorialsVIEW ALL [1]

Solution Differentiate ( Sin X ) Cos X ? Concept: Simple Problems on Applications of Derivatives.
S
View in app×