Question

Differentiate

${\left(\sin x\right)}^{\cos x}$

Answer 1

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