Question:

<b>Composite Function Derivative</b><br> $f(x) = x+2 \Rightarrow f'(x) = 1$ (constant).<br> $f(f(x)) = f(x+2) = (x+2)+2 = x+4$, so $\dfrac{d}{dx}f(f(x)) = 1$.<br> Equivalently: $f'(f(x))\cdot f'(x) = 1\cdot 1 = 1$ for all $x$, including $x=4$.<br> <b>Answer: 1</b><br> <b>Key concept:</b> If $f$ is linear ($f'=$const), then $f'(f(x))=f'(x)=\text{const}$ everywhere.<br> <b>Trap:</b> Confusing $f'(f(x))$ with $(f\circ f)'(x)=f'(f(x))\cdot f'(x)$. The question asks specifically for $f'(f(x))$, not the chain-rule product.