To solve this problem, let's go through each part step by step.

Step 1: Identify $n$ in the function $h(x) = (8x^3 - 16e^x)^5$

In this expression, the outer function is raised to the power of 5, meaning we use the power rule for differentiation with $n = 5$.

So, $n = 5$

Step 2: Apply the Power Rule for Differentiation

The power rule for differentiation states: [ \frac{d}{dx}[f(x)]^n = n[f(x)]^{n-1} \cdot f'(x) ] In this case:

• $f(x) = 8x^3 - 16e^x$
• $n = 5$

To differentiate $h(x)$, we apply the rule as follows:

$h'(x) = 5 \cdot (8x^3 - 16e^x)^4 \cdot \frac{d}{dx}(8x^3 - 16e^x)$

Step 3: Differentiate $f(x) = 8x^3 - 16e^x$

Now, we find $f'(x)$: $f'(x) = \frac{d}{dx}(8x^3) - \frac{d}{dx}(16e^x)$ $f'(x) = 24x^2 - 16e^x$

Step 4: Substitute and Simplify

Now, plug $f(x)$, $n$, and $f'(x)$ back into our expression for $h'(x)$: $h'(x) = 5 \cdot (8x^3 - 16e^x)^4 \cdot (24x^2 - 16e^x)$

Thus, the derivative $h'(x)$ is: $h'(x) = 5(8x^3 - 16e^x)^4(24x^2 - 16e^x)$

Would you like more detailed steps or further clarification on any part?

Related Questions

1. How do you apply the chain rule in calculus?
2. What is the derivative of an exponential function?
3. How would this problem change if $h(x)$ were raised to a different power, say 3 instead of 5?
4. How do you find the derivative of a function involving a product of two terms?
5. What are the common rules of differentiation for polynomials and exponentials?

Tip

Always remember to apply the chain rule when differentiating composite functions, as it helps manage functions nested within others.