The problem asks to use the chain rule to find the derivative of the function $f(x) = 3e^{7x^5 + 8x^6}$.

Solution:

The chain rule states that if a function is a composition of two or more functions, the derivative of the composition is the derivative of the outer function times the derivative of the inner function.

Here, the function has the form $f(x) = 3e^{u(x)}$, where $u(x) = 7x^5 + 8x^6$.

1. Differentiate the outer function: The derivative of $3e^{u(x)}$ with respect to $u(x)$ is: $\frac{d}{du} \left( 3e^{u(x)} \right) = 3e^{u(x)}$

2. Differentiate the inner function $u(x) = 7x^5 + 8x^6$: $\frac{d}{dx}(u(x)) = \frac{d}{dx}(7x^5 + 8x^6) = 35x^4 + 48x^5$

3. Apply the chain rule: Now, apply the chain rule: $f'(x) = 3e^{u(x)} \cdot \frac{d}{dx}(u(x)) = 3e^{7x^5 + 8x^6} \cdot (35x^4 + 48x^5)$

Thus, the derivative of $f(x)$ is: $f'(x) = 3e^{7x^5 + 8x^6}(35x^4 + 48x^5)$

Do you want further details or explanations?

Here are 5 related questions to expand on this topic:

1. How do you apply the chain rule when there are multiple nested functions?
2. What is the derivative of $e^{x^2}$?
3. How does the product rule differ from the chain rule?
4. Can you explain how the chain rule applies when differentiating $\sin(x^2)$?
5. What is the second derivative of $f(x)$ in this case?

Tip: When applying the chain rule, always identify the "outer" and "inner" functions to simplify the differentiation process.