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The given function is \( f(x) = 4x^5 \), and you are asked to find the second derivative at the point \( x = -3 \).

Let's solve this step-by-step:

### 1. First Derivative \( f'(x) \):
Use the power rule, which states that \( \frac{d}{dx}[x^n] = nx^{n-1} \).

\[
f'(x) = \frac{d}{dx}(4x^5) = 4 \cdot 5x^{5-1} = 20x^4
\]

### 2. Second Derivative \( f''(x) \):
Now, take the derivative of \( f'(x) = 20x^4 \).

\[
f''(x) = \frac{d}{dx}(20x^4) = 20 \cdot 4x^{4-1} = 80x^3
\]

### 3. Evaluate at \( x = -3 \):

Substitute \( x = -3 \) into \( f''(x) \):

\[
f''(-3) = 80(-3)^3 = 80(-27) = -2160
\]

So, the second derivative at \( x = -3 \) is \( \boxed{-2160} \).

Would you like more details or have any questions?

Here are some related questions:
1. How is the power rule applied to derivatives?
2. What is the general method to find higher-order derivatives?
3. How does the behavior of a polynomial function relate to its second derivative?
4. Can the second derivative be interpreted geometrically? 
5. What happens if you find higher-order derivatives for \( f(x) = 4x^5 \)?

**Tip:** For polynomials, the degree of the function decreases with each derivative.

What is the second derivative of f(x) = 4x^5 at the point x = -3?

Learn how to compute the second derivative of the polynomial function f(x) = 4x^5 and evaluate it at the point x = -3. This problem applies the power rule of differentiation step-by-step. Suitable for students in Grades 11-12 or early calculus courses.

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