Question

If $y=x^{\sin x}+(\sin x{)}^x$ then $\frac{dy}{dx}$ at $x=\frac{\pi }{2}$ is

• A. $\frac{4}{\pi }$
• B. $1$
• C. $\pi \log \frac{\pi }{2}$
• D. $\frac{{\pi }^2}{2}$

Answer 1

Answer: (B)

$y=x^{\sin x}+(\sin x{)}^x$
$\frac{dy}{dx}=\left[x^{\sin x}\right]\left[\frac{\sin x}{x}+\cos x\cdot \log x\right]+$
$(\sin x{)}^x[x\cos x+\log \sin x]$
$x=\frac{\pi }{2}$
$=\frac{\pi }{2}\left[\frac{2}{\pi }\right]+1[0+0]=1$