To solve for the derivative of $f(x) = (7x^2 + 4x + 6)^{-2}$, we will apply the chain rule.

Let: $u = 7x^2 + 4x + 6$ Then: $f(x) = u^{-2}$

Step 1: Differentiate $f(x) = u^{-2}$ with respect to $u$:

$\frac{d}{du}(u^{-2}) = -2u^{-3}$

Step 2: Differentiate $u = 7x^2 + 4x + 6$ with respect to $x$:

$\frac{du}{dx} = 14x + 4 = 2(7x + 2)$

Step 3: Apply the chain rule

Using $f'(x) = \frac{df}{du} \cdot \frac{du}{dx}$: $f'(x) = -2u^{-3} \cdot (2)(7x + 2)$ Substituting back $u = 7x^2 + 4x + 6$: $f'(x) = -4(7x + 2)(7x^2 + 4x + 6)^{-3}$

Final Answer:

The correct choice is: $f'(x) = -4(7x + 2)(7x^2 + 4x + 6)^{-3}$

Would you like further details or have any questions?

Here are five questions to expand on this topic:

1. How does the chain rule apply in different contexts of calculus?
2. What is the importance of correctly identifying inner and outer functions in the chain rule?
3. How does this derivative process change if the exponent were different?
4. Could we apply the product rule in any part of this problem, and why or why not?
5. What are some common mistakes to avoid when using the chain rule?

Tip: When applying the chain rule, always clearly separate the steps of differentiating the outer function and the inner function before combining them.