To find the derivative of the function $f(x) = (5x + 2)^{-2}$, we will use the chain rule of differentiation.

Step-by-step process:

1. The function $f(x) = (5x + 2)^{-2}$ is a composite function, where the outer function is $u^{-2}$ and the inner function is $u = 5x + 2$.

2. First, differentiate the outer function $u^{-2}$ with respect to $u$, using the power rule: $\frac{d}{du}(u^{-2}) = -2u^{-3}$

3. Now differentiate the inner function $u = 5x + 2$ with respect to $x$: $\frac{d}{dx}(5x + 2) = 5$

4. Now apply the chain rule, multiplying the derivative of the outer function by the derivative of the inner function: $f'(x) = -2(5x + 2)^{-3} \cdot 5$

5. Simplify the expression: $f'(x) = -10(5x + 2)^{-3}$

Final answer:

$f'(x) = -10(5x + 2)^{-3}$

Would you like further details or clarifications? Let me know if you have more questions!

Here are 5 related questions to further explore:

1. How is the chain rule applied in finding the derivative of composite functions?
2. What is the power rule, and how is it used in differentiation?
3. How would you differentiate a more complex function like $(3x^2 + 7x - 4)^{-3}$?
4. What is the product rule, and when is it used in derivatives?
5. How would you find the second derivative $f''(x)$ for this function?

Tip: When applying the chain rule, always start by identifying the outer function and inner function to break the problem into simpler steps.