To differentiate \( y = (x^{-2} + x)^{-3} \), we can apply the chain rule and power rule.

### Step 1: Apply the chain rule
The general form of the chain rule is:

\[
\frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x)
\]

Here, we have \( f(u) = u^{-3} \) and \( g(x) = x^{-2} + x \).

### Step 2: Differentiate the outer function
Differentiate \( f(u) = u^{-3} \) with respect to \( u \):

\[
f'(u) = -3u^{-4}
\]

So, the derivative of the outer function becomes:

\[
\frac{d}{dx} \left( (x^{-2} + x)^{-3} \right) = -3(x^{-2} + x)^{-4} \cdot \frac{d}{dx}(x^{-2} + x)
\]

### Step 3: Differentiate the inner function
Now differentiate \( g(x) = x^{-2} + x \) with respect to \( x \):

\[
g'(x) = \frac{d}{dx}(x^{-2}) + \frac{d}{dx}(x) = -2x^{-3} + 1
\]

### Step 4: Combine the results
Now combine everything to get the final derivative:

\[
\frac{dy}{dx} = -3(x^{-2} + x)^{-4} \cdot (-2x^{-3} + 1)
\]

Thus, the derivative of \( y = (x^{-2} + x)^{-3} \) is:

\[
\frac{dy}{dx} = -3(x^{-2} + x)^{-4} \cdot (-2x^{-3} + 1)
\]

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Would you like more details or have any questions about this process? Here are some related questions you might find interesting:

1. How would the derivative change if the exponent was different?
2. Can you apply the same method for other types of nested functions?
3. What happens if the inner function is a simple polynomial, like \( x^2 + 3x \)?
4. How do you simplify derivatives like this one when you get the result?
5. Could you use the product rule here instead of the chain rule? 

**Tip**: Always check if you can simplify the expression before differentiating—sometimes, this can save you from dealing with more complicated terms.

Learn how to find the derivative of y = (x^-2 + x)^-3 using the chain rule and power rule in this detailed step-by-step solution. The process involves differentiating both the outer and inner functions. Suitable for high school students in Grades 11-12 studying calculus.

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