Question

the following table lists the values of functions g and h, and of their derivatives, g and h, for x = 0.

x	g(x)	h(x)	g(x)	h(x)

let function f be defined as f(x)=g(x)·h(x).
f(0)=□

Explanation:

Step1: Recall product - rule for differentiation

The product - rule states that if $F(x)=g(x)\cdot h(x)$, then $F^{\prime}(x)=g^{\prime}(x)h(x)+g(x)h^{\prime}(x)$.

Step2: Evaluate $F^{\prime}(0)$

Substitute $x = 0$ into the product - rule formula. We know from the table that when $x = 0$, $g(0)=-3$, $h(0)=-1$, $g^{\prime}(0)=5$, and $h^{\prime}(0)=3$.
So, $F^{\prime}(0)=g^{\prime}(0)h(0)+g(0)h^{\prime}(0)$.
$F^{\prime}(0)=(5)\times(-1)+(-3)\times(3)$.

Step3: Perform the arithmetic operations

First, calculate $(5)\times(-1)=-5$ and $(-3)\times(3)=-9$.
Then, $F^{\prime}(0)=-5+( - 9)=-14$.

Answer:

$-14$