To solve the derivative problem in the image, we are tasked with finding $f'(x)$, where:

$f(x) = e^{9x}$

Solution:

We apply the chain rule to differentiate the function $f(x) = e^{9x}$.

1. The derivative of $e^{u}$ (where $u$ is a function of $x$) is given by: $\frac{d}{dx} e^u = e^u \cdot \frac{du}{dx}$

2. In this case, $u = 9x$, and the derivative of $9x$ with respect to $x$ is $9$.

Thus, the derivative of $f(x) = e^{9x}$ is:

$f'(x) = e^{9x} \cdot 9 = 9e^{9x}$

Final Answer:

$f'(x) = 9e^{9x}$

Would you like further details or have any questions?

Here are five related questions to expand your understanding:

1. How would the derivative change if $f(x) = e^{kx}$ where $k$ is a constant?
2. What is the second derivative of $f(x) = e^{9x}$?
3. How does the chain rule apply when differentiating more complex functions?
4. Can you apply the same method to differentiate $f(x) = e^{9x^2}$?
5. What is the integral of $f(x) = e^{9x}$?

Tip: When using the chain rule, always identify the inner function and its derivative first.