Question

Differentiate w.r.t. x :

$f\left(x\right)=\sqrt{\sin \left(\cos x\right)}$

Answer 1

Given $f\left(x\right)=\sqrt{\sin \left(\cos x\right)}$

Differentiating it w.r.t $x$

Apply chain rule

$\left[u\right(x{)}^n{]}^{\prime }=n.u(x{)}^{n-1}.u^{\prime }(x)$

$f^{\prime }\left(x\right)=\frac{1}{2}{\sin }^{\frac{1}{2}-1}\left(\cos x\right).\frac{d}{d}\left[\sin \right(\cos x\left)\right]$

Apply differentiation rule

$\left[\sin \right(u\left(x\right)){]}^{\prime }=[\cos \left(u\right(x\left)\right)].u(x{)}^{\prime }$

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

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