Question

(1) which is the correct formula for finding the derivative of the product of two functions?
(fg) = f+g
(fg) = fg+gf
(fg) = fg
(2) use the correct formula above to find the derivative of the function f(x)=x^8e^x.

Explanation:

Step1: Identify functions

Let $f(x)=x^{8}$ and $g(x)=e^{x}$.

Step2: Find derivatives of individual functions

The derivative of $f(x)=x^{8}$ using the power - rule $(x^{n})'=nx^{n - 1}$ is $f'(x)=8x^{7}$. The derivative of $g(x)=e^{x}$ is $g'(x)=e^{x}$.

Step3: Apply product - rule

The product - rule $(fg)'=fg'+gf'$. Substitute $f(x)=x^{8}$, $f'(x)=8x^{7}$, $g(x)=e^{x}$, and $g'(x)=e^{x}$ into the formula. We get $(x^{8}e^{x})'=x^{8}e^{x}+8x^{7}e^{x}=x^{7}e^{x}(x + 8)$.

Answer:

(1) $(fg)'=fg'+gf'$
(2) $x^{7}e^{x}(x + 8)$