Question

What is the derivative of $\sin \left(\sin x\right)$?

• A. $\cos \left(\cos x\right)$
• B. $\cos \left(\sin x\right)$
• C. $\cos \left(\sin x\right)\cos x$
• D. $\cos \left(\cos x\right)\cos x$

Answer 1

The correct option is C $\cos \left(\sin x\right)\cos x$

Let the given function is $f\left(x\right)=\sin \left(\sin x\right)$

Let $\sin x=t$

Differentiation on both sides, we get

$dt=\cos x$

So, $f\left(x\right)=\sin \left(t\right)$

Differentiating on both sides, we get

$f^{\prime }\left(x\right)=\cos t.dt$

We know $dt=\cos x$

$\therefore f^{\prime }\left(x\right)=\cos \left(t\right).\cos x=\cos \left(\sin x\right).\cos x$