Question

Differentiate \[x^{\sin x}\]  ?

Answer 1

\[\text{ Let y }= x^{\sin x} . . . \left( i \right)\]

Taking log on both sides,

\[\log y = \log x^{\sin x} \]
\[ \Rightarrow \log y = \sin x \log x \left[ \because \log a^b = b \log a \right]\]

Differentiating with respect to x, we get,

\[\Rightarrow \frac{1}{y}\frac{dy}{dx} = \sin x\frac{d}{dx}\log x + \log x\frac{d}{dx}\sin x \left[ \text{ using product rule } \right]\]
\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = \sin x\left( \frac{1}{x} \right) + \log x\left( \cos x \right)\]
\[ \Rightarrow \frac{dy}{dx} = y\left[ \frac{\sin x}{x} + \left( \log x \right)\left( \cos x \right) \right]\]

\[\Rightarrow \frac{dy}{dx} = x^{\sin x} \left[ \frac{\sin x}{x} + \left( \log x \right)\left( \cos x \right) \right]\]   [From (i)]