If $y=(\sin x{)}^{\tan x}$, find $\frac{dy}{dx}$ at $x=\frac{\pi }{4}$.

Solution

1. Take the natural logarithm: $\ln y=\tan x\ln (\sin x)$.
2. Differentiate with respect to $x$: $\frac{1}{y}\frac{dy}{dx}={\sec }^{2}x\ln (\sin x)+\tan x\frac{\cos x}{\sin x}$.
3. Simplify: $\frac{dy}{dx}=y[{\sec }^{2}x\ln (\sin x)+1]$.
4. At $x=\frac{\pi }{4}$, $$\begin{array}{rl}y& =(1/\sqrt{2}{)}^1\\ & =1/\sqrt{2}\end{array}$$, ${\sec }^2(\pi /4)=2$, $$\begin{array}{rl}\ln (\sin (\pi /4))& =\ln (1/\sqrt{2})\\ & =-\frac{1}{2}\ln 2\end{array}$$.
5. Hence the answer is (A).