https://file.aiquickdraw.com/mathbot/file/1727645945616ui978k76.jpg

Let's work through the two questions visible in your image.

### Question 3:
Find all values of \( x = c \) so that the tangent line to the graph of \( f(x) \) at \( (c, f(c)) \) will be horizontal.

Given the function:
\[ f(x) = (x^2 + 5x)^4 \]

A horizontal tangent line means that the derivative of the function, \( f'(x) \), is equal to zero. So, we need to:
1. Find \( f'(x) \),
2. Set \( f'(x) = 0 \),
3. Solve for \( x \).

#### Step 1: Find \( f'(x) \)
Using the chain rule:
\[
f(x) = (x^2 + 5x)^4
\]
Let \( u = x^2 + 5x \), so \( f(x) = u^4 \).

The derivative of \( u^4 \) is \( 4u^3 \), and the derivative of \( u = x^2 + 5x \) is \( 2x + 5 \). Thus:
\[
f'(x) = 4(x^2 + 5x)^3 \cdot (2x + 5)
\]

#### Step 2: Set \( f'(x) = 0 \)
To find where the tangent is horizontal, set:
\[
4(x^2 + 5x)^3 \cdot (2x + 5) = 0
\]
This equation holds if either:
1. \( (x^2 + 5x) = 0 \), or
2. \( (2x + 5) = 0 \).

Solving each case:
- \( x^2 + 5x = 0 \) gives \( x(x + 5) = 0 \), so \( x = 0 \) or \( x = -5 \).
- \( 2x + 5 = 0 \) gives \( x = -\frac{5}{2} \).

#### Step 3: Solutions
The values of \( x \) that satisfy the condition are:
\[
x = 0, \quad x = -5, \quad x = -\frac{5}{2}
\]

So, the correct answer is: \( x = 0, x = -5, x = -\frac{5}{2} \).

### Question 4:
Differentiate the given function and simplify your answer.

The function is:
\[
f(x) = (x + 3)^3 (2x - 1)^5
\]
We need to apply the product rule and the chain rule.

The product rule states:
\[
(fg)' = f'g + fg'
\]

Let \( u = (x + 3)^3 \) and \( v = (2x - 1)^5 \).

1. Differentiate \( u \):
\[
u' = 3(x + 3)^2
\]

2. Differentiate \( v \):
\[
v' = 5(2x - 1)^4 \cdot 2 = 10(2x - 1)^4
\]

Now apply the product rule:
\[
f'(x) = u'v + uv'
\]
\[
f'(x) = 3(x + 3)^2 (2x - 1)^5 + (x + 3)^3 \cdot 10(2x - 1)^4
\]

This is the derivative of the given function, and it can be simplified further if necessary.

Would you like more details or clarification on any step?

### Follow-up Questions:
1. How do you apply the chain rule to more complex functions?
2. What are the conditions for a horizontal tangent on a graph?
3. Can you explain the product rule in different contexts?
4. How do you simplify derivatives with multiple terms?
5. When is it useful to factor derivatives after differentiation?

**Tip:** When working with product and chain rules together, make sure to carefully apply each rule step by step to avoid errors.

Find all values of x = c so that the tangent line to the graph of f(x) at (c, f(c)) will be horizontal.
f(x) = (x^2 + 5x)^4

Learn how to find all values of x = c where the tangent line to the graph of f(x) = (x^2 + 5x)^4 is horizontal. This problem involves key concepts like derivatives, the chain rule, and horizontal tangents, suitable for college-level calculus students.

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