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The problem provided asks to express the given function in the composite form \( f(g(x)) \), identify the inner function \( u = g(x) \) and the outer function \( y = f(u) \), and find the derivative \( \frac{dy}{dx} \).

The function given is:
\[
y = e^{9 \sqrt{x}}
\]

### Step 1: Identifying the Composite Function

We need to rewrite \( y = e^{9 \sqrt{x}} \) in the form \( f(g(x)) \):

- **Inner function** \( g(x) \): The part inside the exponent, \( g(x) = 9 \sqrt{x} \).
- **Outer function** \( f(u) \): The exponential function applied to \( g(x) \), so \( f(u) = e^u \), where \( u = 9 \sqrt{x} \).

Thus, we can write:
\[
f(g(x)) = e^{9 \sqrt{x}}
\]

### Step 2: Find the Derivative \( \frac{dy}{dx} \)

To find \( \frac{dy}{dx} \), we'll use the chain rule, since we have a composite function \( y = f(g(x)) \).

1. **Derivative of the outer function** \( f(u) = e^u \):
   \[
   \frac{d}{du} e^u = e^u
   \]

2. **Derivative of the inner function** \( g(x) = 9 \sqrt{x} \):
   \[
   g(x) = 9x^{1/2}, \quad \frac{dg(x)}{dx} = 9 \times \frac{1}{2} x^{-1/2} = \frac{9}{2 \sqrt{x}}
   \]

Now, applying the chain rule:
\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = e^{9 \sqrt{x}} \cdot \frac{9}{2 \sqrt{x}}
\]

Thus, the derivative is:
\[
\frac{dy}{dx} = \frac{9 e^{9 \sqrt{x}}}{2 \sqrt{x}}
\]

### Final Answer:

- Composite function: \( f(g(x)) = e^{9 \sqrt{x}} \)
- Derivative: \( \frac{dy}{dx} = \frac{9 e^{9 \sqrt{x}}}{2 \sqrt{x}} \)

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Would you like any further details or have any questions?

Here are 5 related questions:
1. How would the process change if the exponent were a different function of \(x\)?
2. How can we apply the chain rule to more complex functions?
3. What would be the derivative if \( y = e^{ax} \) for any constant \( a \)?
4. What other techniques besides the chain rule are useful for derivatives of composite functions?
5. How do we find second derivatives of composite functions like this one?

**Tip**: When dealing with composite functions, always break them down into inner and outer components to simplify differentiation using the chain rule.

Write the composite function in the form f(g(x)) for y = e^{9√x}. Identify the inner function u = g(x) and the outer function y = f(u). Then, find the derivative dy/dx.

Learn how to express y = e^(9√x) as a composite function f(g(x)) and calculate its derivative using the chain rule. This problem demonstrates key calculus concepts, including exponential functions and applying the chain rule.

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