To find the derivative of $(-2x^2 + 5)(2x^2 + 5)^{-2}$, we need to apply the product rule and the chain rule.

Step 1: Applying the product rule

The product rule states: $\frac{d}{dx}[u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x)$ Here, we let:

• $u(x) = -2x^2 + 5$
• $v(x) = (2x^2 + 5)^{-2}$

Step 2: Differentiating each function

• To find $u'(x)$, differentiate $u(x) = -2x^2 + 5$: $u'(x) = -4x$

• To differentiate $v(x) = (2x^2 + 5)^{-2}$, we need to use the chain rule: $v'(x) = \frac{d}{dx} \left[(2x^2 + 5)^{-2}\right] = -2(2x^2 + 5)^{-3} \cdot \frac{d}{dx}(2x^2 + 5)$ The derivative of $2x^2 + 5$ is $4x$, so: $v'(x) = -2(2x^2 + 5)^{-3} \cdot 4x = -8x(2x^2 + 5)^{-3}$

Step 3: Putting it all together

Now, using the product rule: $\frac{d}{dx}\left[(-2x^2 + 5)(2x^2 + 5)^{-2}\right] = u'(x) \cdot v(x) + u(x) \cdot v'(x)$ Substituting the derivatives: $= (-4x)(2x^2 + 5)^{-2} + (-2x^2 + 5)(-8x)(2x^2 + 5)^{-3}$ Simplifying: $= -4x(2x^2 + 5)^{-2} + 16x(-2x^2 + 5)(2x^2 + 5)^{-3}$

This is the derivative of the given function.

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Do you want a more detailed explanation of any part of this process?

Here are 5 related questions:

1. What is the product rule, and how does it apply in this case?
2. How does the chain rule help in differentiating composite functions like $(2x^2 + 5)^{-2}$?
3. How do you simplify expressions with exponents like $(2x^2 + 5)^{-2}$?
4. How can you verify this derivative using a graph or a numerical method?
5. What are other methods for differentiating more complex product functions?

Tip: When differentiating functions involving products or powers, break down the components and apply the product rule and chain rule step-by-step to avoid mistakes.