Question:

Let $f(x)$ and $g(x)$ be differentiable functions, and let $h(x) = f(x) \cdot g(x)$. Suppose $f'(x) = 3x^2 - 2$ and $g'(x) = \frac{4}{x}$ for all $x \neq 0$.

a) Using the product rule, find an expression for $h'(x)$ in terms of $f(x)$, $f'(x)$, $g(x)$, and $g'(x)$.

b) Evaluate $h'(2)$.

Answer:

a) According to the product rule, the derivative of the product of two functions is given by:

Substituting the given functions, we have:

Therefore, we can express $h'(x)$ in terms of $f(x)$, $f'(x)$, $g(x)$, and $g'(x)$ as:

b) To evaluate $h'(2)$, we substitute $x = 2$ into the expression for $h'(x)$ from part a:

Using the given information that $f'(x) = 3x^2 - 2$ and $g'(x) = \frac{4}{x}$, we can find $f(2)$ and $g(2)$.

Since $f'(x) = 3x^2 - 2$, we can integrate $f'(x)$ to find $f(x)$. Integrating, we get:

where $C$ is the constant of integration.

To find the constant of integration, we need additional information. Since no other information is given, we assume $C = 0$ for simplicity. Therefore, $f(x) = x^3 - 2x$.

Next, we find $g(2)$ using the given information $g'(x) = \frac{4}{x}$. Integrating, we get:

Again, assuming $C = 0$ for simplicity, we have $g(x) = 4\ln|x|$.

Substituting the found values into $h'(2)$, we get:

Therefore, $h'(2) = 8 + 80\ln|2|$.

The final result is $h'(2) = 8 + 80\ln|2|$.