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To solve this problem, let's break down the composite function \( y = e^{9\sqrt{x}} \) into inner and outer functions as \( y = f(g(x)) \).

### Step 1: Identify \( f(u) \) and \( g(x) \)
We want to express \( y = e^{9\sqrt{x}} \) in terms of two functions:
1. **Inner function** \( g(x) \): Let \( g(x) = 9\sqrt{x} \).
2. **Outer function** \( f(u) \): Then, \( f(u) = e^u \).

So, we can write:
\[
f(u) = e^u \quad \text{and} \quad g(x) = 9\sqrt{x}.
\]

This makes \( y = f(g(x)) \).

### Step 2: Find the derivative \( \frac{dy}{dx} \)
To find \( \frac{dy}{dx} \), we’ll use the chain rule:
\[
\frac{dy}{dx} = f'(g(x)) \cdot g'(x).
\]

#### Step 2a: Find \( f'(u) \)
Since \( f(u) = e^u \), the derivative \( f'(u) = e^u \).

#### Step 2b: Find \( g'(x) \)
Since \( g(x) = 9\sqrt{x} = 9x^{1/2} \), we differentiate with respect to \( x \):
\[
g'(x) = 9 \cdot \frac{1}{2} x^{-\frac{1}{2}} = \frac{9}{2\sqrt{x}}.
\]

#### Step 2c: Substitute and simplify
Now, substitute \( f'(g(x)) = e^{9\sqrt{x}} \) and \( g'(x) = \frac{9}{2\sqrt{x}} \):
\[
\frac{dy}{dx} = e^{9\sqrt{x}} \cdot \frac{9}{2\sqrt{x}} = \frac{9e^{9\sqrt{x}}}{2\sqrt{x}}.
\]

### Final Answer:
1. \( f(u) = e^u \) and \( g(x) = 9\sqrt{x} \).
2. \( \frac{dy}{dx} = \frac{9e^{9\sqrt{x}}}{2\sqrt{x}} \).

Would you like further details on any of these steps, or do you have questions?

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Here are 5 related questions to expand on this topic:
1. How would you apply the chain rule to a different composite function, such as \( y = \sin(3x^2) \)?
2. What if the exponent in the outer function were different, like \( y = e^{5\sqrt{x}} \)? How would that affect the derivative?
3. Can you identify \( f(u) \) and \( g(x) \) in a similar function, \( y = \ln(x^2 + 1) \)?
4. How does the chain rule differ from the product rule in differentiation?
5. How would the derivative change if \( g(x) = 9x^{3/2} \) instead?

**Tip:** Always identify inner and outer functions clearly in composite functions; this simplifies applying the chain rule effectively.

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

Explore how to write the composite function y = e^(9√x) in terms of inner and outer functions, then apply the chain rule to find dy/dx. Suitable for Grades 11-12.

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