If f(4)=−2, f′(4)=5, g(4)=14, and g′(4)=−3, find the following function value.

(8fg)′(4)

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!

If f(4)=−2, f′(4)=5, g(4)=14, and g′(4)=−3, find the following function value: (8fg)′(4).

Learn how to solve the differentiation problem (8fg)′(4) using the product rule. With given values for f(4), f'(4), g(4), and g'(4), this problem provides a step-by-step breakdown, suitable for students in Grades 11-12, covering key calculus concepts.

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