Let k(x) = f(x)g(x). Use the graphs above to find the following derivatives.k '(1) = k '(3) =

To find the derivatives $k'(1)$ and $k'(3)$ where $k(x) = f(x)g(x)$, we can apply the product rule for differentiation, which states:

$k'(x) = f'(x)g(x) + f(x)g'(x)$

1. Break Down the Problem

We need to compute the derivative $k'(x)$ at specific points $x = 1$ and $x = 3$. We'll express $k'(1)$ and $k'(3)$ using the product rule.

2. Relevant Concepts

We will identify the values of $f(1), g(1), f'(1), g'(1)$ for $k'(1)$ and the values of $f(3), g(3), f'(3), g'(3)$ for $k'(3)$ using the graphs of $f(x)$ and $g(x)$.

3. Analysis and Detail

Assuming we have access to the graphs of $f(x)$ and $g(x)$:

1. For $k'(1)$:

   • Find $f(1)$ and $g(1)$ from the graphs.
   • Find $f'(1)$ and $g'(1)$ from the graphs (this is the slope of the tangent line at $x = 1$).
   • Apply the product rule: $k'(1) = f'(1)g(1) + f(1)g'(1)$

2. For $k'(3)$:

   • Find $f(3)$ and $g(3)$ from the graphs.
   • Find $f'(3)$ and $g'(3)$ from the graphs.
   • Apply the product rule: $k'(3) = f'(3)g(3) + f(3)g'(3)$

4. Verify and Summarize

After calculating $k'(1)$ and $k'(3)$ with the above formulas using the values obtained from the graphs, we can summarize the findings.

Final Answer

• $k'(1) = \text{(calculated value)}$
• $k'(3) = \text{(calculated value)}$

Note: To complete the above values, specific details from the graphs of $f(x)$ and $g(x)$ are required.