Problem
The following table lists the values of functions $g$ and $h$, and of their derivatives, $g^{\prime}$ and $h^{\prime}$, for the $x$-values -2 and 3 . \begin{tabular}{ccccc} $x$ & $g(x)$ & $h(x)$ & $g^{\prime}(x)$ & $h^{\prime}(x)$ \\ -2 & 0 & 3 & 1 & -5 \\ 3 & 5 & 3 & 1 & 5 \end{tabular} Evaluate $\frac{d}{d x}[g(h(x))]$ at $x=-2$
Solution
Step 1 ：Let \(f(x) = g(h(x))\), then the derivative \(f'(x)\) is given by \(f'(x) = g'(h(x)) \cdot h'(x)\) using the chain rule.
Step 2 ：Substitute \(x=-2\) into the equation to get \(f'(-2) = g'(h(-2)) \cdot h'(-2)\).
Step 3 ：From the table, we know that \(h(-2) = 3\) and \(h'(-2) = -5\). So, substitute these values into the equation to get \(f'(-2) = g'(3) \cdot -5\).
Step 4 ：Again from the table, we know that \(g'(3) = 1\). So, substitute this value into the equation to get \(f'(-2) = 1 \cdot -5 = -5\).
Step 5 ：\(\boxed{f'(-2) = -5}\) is the derivative of \(g(h(x))\) at \(x=-2\).
