To find the derivative of the function $y = \frac{1}{(5x + 2)^3}$, we can apply the chain rule.

Step 1: Rewrite the function

First, rewrite the function as a power function: $y = (5x + 2)^{-3}$

Step 2: Apply the chain rule

The chain rule states that if $y = [u(x)]^n$, then: $\frac{dy}{dx} = n \cdot [u(x)]^{n-1} \cdot \frac{du}{dx}$ In this case, $u(x) = 5x + 2$ and $n = -3$.

Step 3: Differentiate

1. Differentiate the outer function: $\frac{dy}{du} = -3 \cdot (5x + 2)^{-4}$
2. Differentiate the inner function $u(x) = 5x + 2$: $\frac{du}{dx} = 5$

Step 4: Combine the results

Now, multiply these results to get the derivative: $\frac{dy}{dx} = -3 \cdot (5x + 2)^{-4} \cdot 5$

Final Answer

$\frac{dy}{dx} = -15 \cdot (5x + 2)^{-4}$ Or equivalently: $\frac{dy}{dx} = \frac{-15}{(5x + 2)^4}$

Would you like more details or have any questions?

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Here are 5 related questions you might find interesting:

1. What is the derivative of $y = \frac{1}{(3x + 1)^2}$?
2. How does the chain rule apply to trigonometric functions?
3. Can you find the second derivative of $y = \frac{1}{(5x + 2)^3}$?
4. What is the integral of $y = (5x + 2)^{-3}$?
5. How would the derivative change if $y = \frac{1}{(5x^2 + 2)^3}$?

Tip: When differentiating composite functions, always identify the outer and inner functions before applying the chain rule.