https://file.aiquickdraw.com/mathbot/file/1727662336348ytu67jt7.jpg

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?

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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.

Learn how to find the derivative of the exponential function f(x) = e^{9x} using the chain rule. This solution demonstrates the steps for applying the chain rule to differentiate exponential functions, suitable for students in Grades 11-12.

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