Question

the following table lists the values of functions f and g, and of their derivatives, f and g, for x = 1.

x	f(x)	g(x)	f(x)	g(x)

evaluate $\frac{d}{dx}f(x)cdot g(x)$ at x = 1.

Explanation:

Step1: Apply product - rule

The product - rule states that $\frac{d}{dx}[f(x)\cdot g(x)]=f(x)g'(x)+g(x)f'(x)$.

Step2: Substitute $x = 1$

When $x = 1$, we know from the table that $f(1)=3$, $g(1)=2$, $f'(1)=7$, and $g'(1)= - 3$.
Substitute these values into the product - rule formula:
\[

$$\begin{align*} \frac{d}{dx}[f(x)\cdot g(x)]\big|_{x = 1}&=f(1)g'(1)+g(1)f'(1)\\ &=3\times(-3)+2\times7 \end{align*}$$

\]

Step3: Calculate the result

\[

$$\begin{align*} 3\times(-3)+2\times7&=-9 + 14\\ &=5 \end{align*}$$

\]

Answer:

$5$