If $y=(\sin x)^{ an x}$, then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ is equal to

Question

If $y=(\sin x)^{	an x}$, then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ is equal to

Aicharya · Accepted Answer

$y=(\sin x)^{	an x}$ Taking logarithm on both sides, we get $\log y=	an x \cdot \log (\sin x)$ Differentiating w.r.t. $x$, we get $\begin{aligned} & \frac{1}{y} \cdot \frac{\mathrm{~d} y}{\mathrm{~d} x}=	an x \cdot \cot x+\log (\sin x) \cdot \sec ^2 x \ & \Rightarrow \frac{\mathrm{~d} y}{\mathrm{~d} x}=(\sin x)^{	an x}\left[1+\sec ^2 x \log (\sin x)ight] \end{aligned}$

If $y=(\sin x)^{\tan x}$, then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ is equal to

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