the following table lists the value of functions f and g, and their derivatives, f and g, for x = 2.\n| x |…
the following table lists the value of functions f and g, and their derivatives, f and g, for x = 2.\n| x | f(x) | g(x) | f(x) | g(x) |\n|----|----|----|----|----|\n| 2 | 10 | - 3 | - 5 | 4 |\nevaluate $\frac{d}{dx}-2f(x)+5g(x)-9$ at x = 2.
Answer
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