Question

the following table lists the value of functions f and g, and their derivatives, f and g, for x = 2.

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

evaluate $\frac{d}{dx}-2f(x)+5g(x)-9$ at x = 2.

Explanation:

Step1: Apply derivative rules

Use the sum - difference rule of derivatives $\frac{d}{dx}(u + v - w)=\frac{du}{dx}+\frac{dv}{dx}-\frac{dw}{dx}$ and the constant - multiple rule $\frac{d}{dx}(cf(x)) = c\frac{d}{dx}f(x)$. So, $\frac{d}{dx}[-2f(x)+5g(x)-9]=-2\frac{d}{dx}f(x)+5\frac{d}{dx}g(x)-\frac{d}{dx}(9)$.
Since the derivative of a constant is 0, $\frac{d}{dx}(9) = 0$. So we have $\frac{d}{dx}[-2f(x)+5g(x)-9]=-2f'(x)+5g'(x)$.

Step2: Substitute $x = 2$

We are given that when $x = 2$, $f'(2)=-5$ and $g'(2)=4$. Substitute these values into $-2f'(x)+5g'(x)$.
$-2f'(2)+5g'(2)=-2\times(-5)+5\times4$.

Step3: Calculate the result

First, calculate $-2\times(-5)=10$ and $5\times4 = 20$. Then $-2\times(-5)+5\times4=10 + 20=30$.

Answer:

30