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?

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.