Question

Differentiate $f\left(x\right)=x\sin \frac{\pi }{x}$ with respect to $x$.

Answer 1

$f\left(x\right)=x\sin \frac{\pi }{x}$

$f^{\prime }\left(x\right)=\sin \frac{\pi }{x}+\cos \frac{\pi }{x}(-\frac{\pi }{x^2})x$

$\Rightarrow f^{\prime }\left(x\right)=\sin \frac{\pi }{x}-\frac{\pi }{x}\cos \frac{\pi }{x}$

Hence $f^{\prime }\left(x\right)=\sin \frac{\pi }{x}-\frac{\pi }{x}\cos \frac{\pi }{x}$.