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  • Differentiate each of the following w.r.t. x (sin x)^cos x

Differentiate each of the following w.r.t. x
(sin x)^{cos x}

Answers (1)

Given:  (sin x)^{cos x}

To Find: Differentiate w.r.t x

We have  (sin x)^{cos x}

Let y=(sin x)^{cos x}

Now, Taking Log on both sides, we get

Log y = cos x.log(sin x)

Now, Differentiate both side w.r.t. x

\begin{aligned} &\frac{1}{y} \frac{d y}{d x}=\cos x \cdot \frac{d}{d x}(\log (\sin x))+\log (\sin x) \cdot \frac{d}{d x}(\cos x)\\ &\text { By using product rule of differentiation }\\ &\frac{1}{y} \frac{d y}{d x}=\cos x \cdot \frac{1}{\sin x} \cdot \frac{d}{d x}(\sin x)+\log (\sin x) \cdot(-\sin x)\\ &\frac{1}{y} \frac{d y}{d x}=\cos x \cdot \frac{1}{\sin x}(\cos x)+\log (\sin x) \cdot(-\sin x)\\ &\frac{1}{y} \frac{d y}{d x}=\cos x \cdot \cot x-\log (\sin x) \cdot(\sin x)\\ &\frac{d y}{d x}=y[\cos x \cdot \cot x-\log (\sin x) \cdot(\sin x)] \end{aligned}

Substitute the value of y, we get

\begin{array}{l} y^{\prime}=(\sin x)^{\cos x}[\cos x \cdot \cot x-\sin x(\log \sin x)] \\ \text { Hence, } y^{\prime}=(\sin x)^{\cos x}[\cos x \cdot \cot x-\sin x(\log \sin x)] \end{array}

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