Differentiate $x^{sin x}, x > 0$ w.r.t $x$ .

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Solution

Let, $y = x^{s i n x}$

Taking $l o g$ on both sides ,

=> $l n y = l n (s i n x) . x$

Differentiating w.r.t $x$ ,

=> $\frac{1}{y} . d y = [\frac{x}{s i n x} + l n (s i n x)] d x$

=> $\frac{d y}{d x} = y [\frac{x}{s i n x} + l n (s i n x)]$

Hence,

=> $\frac{d y}{d x} = x^{s i n x} [\frac{x}{s i n x} + l n (s i n x)]$

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