To find the derivative of $h(x) = 19(4x^2 - 9x)^5$, we can use the chain rule.

Steps:

1. Let $u = 4x^2 - 9x$, so $h(x) = 19u^5$.
2. Using the chain rule, $h'(x) = 19 \cdot 5u^4 \cdot u'$.
3. Now, find $u'$: $u' = \frac{d}{dx}(4x^2 - 9x) = 8x - 9$
4. Substitute $u$ and $u'$ back into the derivative: $h'(x) = 19 \cdot 5(4x^2 - 9x)^4 \cdot (8x - 9)$

Final Answer:

$h'(x) = 95(4x^2 - 9x)^4 (8x - 9)$

Let me know if you want a breakdown of any step.

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1. What is the chain rule, and how does it apply to derivatives of composite functions?
2. How would you find the second derivative of $h(x)$ here?
3. What happens to the derivative if the power of the expression $(4x^2 - 9x)$ is increased?
4. How would this approach differ if $h(x)$ were a product of two separate functions instead of a power?
5. How does the derivative behave as $x$ approaches infinity?

Tip: The chain rule is often paired with the power rule for functions raised to a power, simplifying complex differentiation