The given function is:

$f(x) = \frac{1}{(4x + 5)^4}$

Step 1: Differentiate $f(x)$

We will use the chain rule for differentiation. Let $u = 4x + 5$, then the function becomes:

$f(x) = u^{-4}$

Now, differentiate:

$\frac{d}{dx} \left( u^{-4} \right) = -4u^{-5} \cdot \frac{du}{dx}$

Since $u = 4x + 5$, we have $\frac{du}{dx} = 4$.

Thus, the derivative of $f(x)$ is:

$f'(x) = -4(4x + 5)^{-5} \cdot 4$

Simplifying:

$f'(x) = -16(4x + 5)^{-5}$

Step 2: Evaluate $f'(-2)$

Now, substitute $x = -2$ into the derivative:

$f'(-2) = -16(4(-2) + 5)^{-5}$

Simplifying inside the parentheses:

$f'(-2) = -16(-8 + 5)^{-5} = -16(-3)^{-5}$

Since $(-3)^{-5} = \frac{1}{(-3)^5} = \frac{1}{-243}$, we get:

$f'(-2) = -16 \times \frac{1}{-243} = \frac{16}{243}$

Final Answers:

• $f'(x) = -16(4x + 5)^{-5}$
• $f'(-2) = \frac{16}{243}$

Would you like further clarification or more details? Here are some related questions:

1. How does the chain rule apply to more complex functions?
2. Can you explain how to handle negative exponents in derivatives?
3. What is the interpretation of the derivative in real-world contexts?
4. How can the chain rule be extended to multivariable functions?
5. Could you show the steps for using a different method, like implicit differentiation?

Tip: Always simplify expressions step-by-step, especially when working with negative exponents or fractions to avoid errors in differentiation.