To find $(8fg)'(4)$, we will use the product rule of differentiation and chain rule where needed.

Given information:

• $f(4) = -2$
• $f'(4) = 5$
• $g(4) = 14$
• $g'(4) = -3$

We are asked to find $(8fg)'(4)$.

Step-by-Step Solution:

The function we are differentiating is $8fg$, so let's apply the product rule to differentiate $fg$:

$\frac{d}{dx}(8fg) = 8 \left( f'(x)g(x) + f(x)g'(x) \right)$

Now, substitute $x = 4$ and plug in the given values:

$(8fg)'(4) = 8 \left( f'(4)g(4) + f(4)g'(4) \right)$

Substitute the known values:

$(8fg)'(4) = 8 \left( 5 \cdot 14 + (-2) \cdot (-3) \right)$

Simplify:

$(8fg)'(4) = 8 \left( 70 + 6 \right) = 8 \times 76 = 608$

Thus, the value of $(8fg)'(4)$ is 608.

Would you like a breakdown of any part of the process? Here are some related questions:

1. How do you apply the product rule for derivatives?
2. What is the chain rule in differentiation?
3. How do you find the derivative of a constant multiplied by a function?
4. Can you differentiate a sum of functions using basic rules?
5. Why do we substitute the given values after differentiating?

Tip: Always check that the product rule is applied correctly when dealing with the derivative of two functions!