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: $\frac{dy}{dx} = \frac{9 e^{9 \sqrt{x}}}{2 \sqrt{x}}$

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.