The following table lists the values of functions $f$ and $g$, and of their derivatives, $f^{\prime}$ and $g^{\prime}$, for the $x$-values 0 and 3 .
\begin{tabular}{ccccc}
$x$ & $f(x)$ & $g(x)$ & $f^{\prime}(x)$ & $g^{\prime}(x)$ \\
\hline 0 & 0 & -3 & 2 & 1 \\
3 & -3 & 0 & -4 & 1
\end{tabular}

Let function $F$ be defined as $F(x)=f(g(x))$. $F^{\prime}(3)=$

Answer

Expert–verified

Hide Steps

Answer

Therefore, $\boxed{F^{\prime}(3) = 2}$.

Steps

Step 1 ：Use the chain rule to find $F^{\prime}(x)$, which gives $F^{\prime}(x) = f^{\prime}(g(x)) \cdot g^{\prime}(x)$.

Step 2 ：Substitute $x=3$ into the equation to get $F^{\prime}(3) = f^{\prime}(g(3)) \cdot g^{\prime}(3)$.

Step 3 ：From the table, we know that $g(3) = 0$ and $g^{\prime}(3) = 1$.

Step 4 ：Substitute these values into the equation to get $F^{\prime}(3) = f^{\prime}(0) \cdot 1$.

Step 5 ：From the table, we know that $f^{\prime}(0) = 2$, so $F^{\prime}(3) = 2 \cdot 1 = 2$.

Step 6 ：Therefore, $\boxed{F^{\prime}(3) = 2}$.