To differentiate the function $f(x) = 7e^{8x}$, we apply the chain rule.

The derivative of an exponential function $e^{u(x)}$ is given by $\frac{d}{dx} e^{u(x)} = e^{u(x)} \cdot u'(x)$, where $u(x)$ is the function inside the exponent.

Here, we have: $f(x) = 7e^{8x}$

Differentiate $f(x)$:

$f'(x) = 7 \cdot \frac{d}{dx} \left( e^{8x} \right)$

Since $\frac{d}{dx} e^{8x} = e^{8x} \cdot 8$, we get:

$f'(x) = 7 \cdot e^{8x} \cdot 8$

Simplifying:

$f'(x) = 56e^{8x}$

So, the derivative is:

$f'(x) = 56e^{8x}$

Would you like more details or have any questions?

Here are 5 related questions:

1. What is the derivative of $e^{kx}$ where $k$ is a constant?
2. How do you apply the chain rule in differentiation?
3. Can you differentiate the function $f(x) = 9e^{5x}$?
4. What is the second derivative of $f(x) = 7e^{8x}$?
5. How do you differentiate exponential functions with respect to $x$?

Tip: Always apply the chain rule when differentiating composite functions like $e^{g(x)}$, where $g(x)$ is any function of $x$.