https://file.aiquickdraw.com/mathbot/file/1726626120485bxuewyvk.jpg

We are tasked with finding \(\frac{dy}{dx}\) for the function given in the image.

The function is:

\[
y = e^{\frac{8}{9}x}
\]

### Step 1: Identify \(y = f(u)\) and \(u = g(x)\)

To rewrite this function in the form \(y = f(u)\) and \(u = g(x)\), we can set:

\[
u = \frac{8}{9}x
\]

This means the function for \(y\) becomes:

\[
y = e^u
\]

Thus, we have \(y = f(u)\) where \(f(u) = e^u\) and \(u = g(x)\) where \(g(x) = \frac{8}{9}x\).

### Step 2: Differentiate using the chain rule

We need to compute \(\frac{dy}{dx}\), and we'll use the chain rule:

\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}
\]

- \(\frac{dy}{du} = \frac{d}{du}(e^u) = e^u\)
- \(\frac{du}{dx} = \frac{d}{dx}\left(\frac{8}{9}x\right) = \frac{8}{9}\)

### Step 3: Combine results

Now, substitute the expressions into the chain rule formula:

\[
\frac{dy}{dx} = e^u \cdot \frac{8}{9}
\]

Substitute \(u = \frac{8}{9}x\) back into the equation:

\[
\frac{dy}{dx} = e^{\frac{8}{9}x} \cdot \frac{8}{9}
\]

Thus, the derivative of the function is:

\[
\frac{dy}{dx} = \frac{8}{9}e^{\frac{8}{9}x}
\]

Would you like further explanation or have any questions?

### Related Questions:
1. How do you apply the chain rule in more complex functions?
2. What is the derivative of a natural exponential function in general?
3. How can the chain rule be extended to higher-order derivatives?
4. How do you apply implicit differentiation when functions are not explicitly written in terms of \(x\)?
5. What is the significance of exponential functions in growth and decay problems?

**Tip:** When applying the chain rule, always differentiate the outer function first and then multiply by the derivative of the inner function.

Write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x. y = e^(8/9 * x)

Learn how to differentiate the exponential function y = e^(8/9 * x) using the chain rule. This problem covers the key concepts of differentiation, including the chain rule and exponential functions, with a step-by-step solution. Suitable for high school students in Grades 10-12.

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