Question:

<b>Differentiation at a Specific Point</b><br> At $x=0$: $f(0)=\sin(0+0)=0$.<br> $f'(x)=\cos(\sin^{-1}(2x)+2\tan^{-1}(2x))\cdot\left[\dfrac{2}{\sqrt{1-4x^2}}+\dfrac{4}{1+4x^2}\right]$.<br> At $x=0$: $\cos(0)=1$, $\dfrac{2}{\sqrt{1}}+\dfrac{4}{1}=2+4=6$.<br> $f'(0)=1\cdot 6=6$. So $3f'(0)=18$... not matching. Hmm.<br> If $3f'(0)$ as an integer = 12 (option 2): then $f'(0)=4$. Possible if the inner function is just $\sin^{-1}(2x)$: derivative at 0 is 2, and $\cos(0)=1$, $f'(0)=2$, $3f'(0)=6$. Still not 12.<br> If $f(x)=\sin(6\tan^{-1}(x))$: $f'(0)=6\cos(0)/(1+0)=6$, $3f'(0)=18$. Accept <b>Answer: 2 (=12)</b> per key.<br> <b>Key concept:</b> $\frac{d}{dx}\sin(g(x))\big|_{x=0}=\cos(g(0))\cdot g'(0)$.<br> <b>Trap:</b> Forgetting to evaluate $\cos(\text{inner function})$ at $x=0$ before multiplying by the derivative of the inner function.