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.

This detailed guide explains how to differentiate the function y = 1/(5x + 2)^3 by applying the chain rule. Perfect for high school students in Grades 11-12, the solution includes step-by-step instructions and key calculus concepts.

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