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To find \( f'(-1) \), we need to compute the derivative of the function \( f(x) = \left( 4x^2 + x - 4 \right)^3 \) and evaluate it at \( x = -1 \).

The function is of the form \( f(x) = \left( g(x) \right)^3 \), where \( g(x) = 4x^2 + x - 4 \). To find \( f'(x) \), we'll apply the chain rule.

1. **Step 1: Apply the chain rule.**

\[
f'(x) = 3 \left( g(x) \right)^2 \cdot g'(x)
\]

2. **Step 2: Compute \( g'(x) \).**

\[
g(x) = 4x^2 + x - 4
\]
\[
g'(x) = 8x + 1
\]

3. **Step 3: Substitute \( g(x) \) and \( g'(x) \) into the expression for \( f'(x) \).**

\[
f'(x) = 3 \left( 4x^2 + x - 4 \right)^2 \cdot (8x + 1)
\]

4. **Step 4: Evaluate at \( x = -1 \).**

First, calculate \( g(-1) \):

\[
g(-1) = 4(-1)^2 + (-1) - 4 = 4 - 1 - 4 = -1
\]

Now calculate \( g'(-1) \):

\[
g'(-1) = 8(-1) + 1 = -8 + 1 = -7
\]

Now, substitute \( x = -1 \) into \( f'(x) \):

\[
f'(-1) = 3 \left( -1 \right)^2 \cdot (-7) = 3 \cdot 1 \cdot (-7) = -21
\]

Thus, \( f'(-1) = -21 \).

---

Would you like more details or have any further questions?

Here are 5 related questions to explore:

1. How does the chain rule apply when differentiating composite functions?
2. What happens if the exponent in the function \( f(x) \) changes to 4 or another value?
3. Can you apply the product rule in place of the chain rule for this function?
4. How would the derivative differ if \( g(x) \) was a different polynomial function?
5. What is the geometric interpretation of finding \( f'(-1) \)?

**Tip:** When using the chain rule, always identify the "outer" and "inner" functions first to simplify the differentiation process.

Find the derivative of the function f(x) = (4x^2 + x - 4)^3 at x = -1.

Learn how to differentiate the function f(x) = (4x^2 + x - 4)^3 using the chain rule. The derivative is evaluated at x = -1, following a step-by-step approach with clear explanations of applying the chain rule.

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