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 $x=-2$

Answer

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Answer

$\boxed{f'(-2) = -5}$ is the derivative of $g(h(x))$ at $x=-2$.

Steps

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 $x=-2$.