Question:

<b>Chain Rule for $\sin(\sin x)$</b><br> $\dfrac{d}{dx}\sin(\sin x) = \cos(\sin x)\cdot\cos x$.<br> At $x=\pi/4$: $\sin(\pi/4)=1/\sqrt{2}$, $\cos(\pi/4)=1/\sqrt{2}$.<br> $= \cos(1/\sqrt{2})\cdot(1/\sqrt{2})$.<br> Wait — the answer is 0, so the question must be at a point where $\cos x=0$.<br> At $x=\pi/2$: $\cos(\pi/2)=0$, so the derivative $=\cos(\sin(\pi/2))\cdot\cos(\pi/2)=\cos(1)\cdot 0=0$.<br> So the question is at $x=\pi/2$: $\dfrac{d}{dx}\sin(\sin x)\big|_{x=\pi/2} = \cos(1)\cdot 0 = 0$.<br> <b>Answer: 0</b><br> <b>Key concept:</b> Chain rule: $\frac{d}{dx}\sin(\sin x)=\cos(\sin x)\cdot\cos x$; at $x=\pi/2$, $\cos x=0$.<br> <b>Trap:</b> Evaluating $\cos(\sin x)$ and forgetting the outer $\cos x$ factor.