Let's solve the problem step-by-step!

Problem Setup:

We are asked to rewrite the function $y = e^{\frac{2}{7}x}$ in the form: $y = f(u) \quad \text{and} \quad u = g(x),$ and then find the derivative $\frac{dy}{dx}$ using the chain rule.

Step 1: Rewrite the function in terms of $u$.

We let: $u = \frac{2}{7}x.$ Then, the original function becomes: $y = e^u.$

Now we have:

• $y = f(u) = e^u$,
• $u = g(x) = \frac{2}{7}x$.

Step 2: Apply the Chain Rule to find $\frac{dy}{dx}$.

Using the chain rule: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}.$

First, compute the individual derivatives:

1. $\frac{dy}{du} = e^u$.
2. $\frac{du}{dx} = \frac{2}{7}$.

So: $\frac{dy}{dx} = e^u \cdot \frac{2}{7}.$

Step 3: Substitute $u = \frac{2}{7}x$ back.

$\frac{dy}{dx} = e^{\frac{2}{7}x} \cdot \frac{2}{7}.$

Final Answer:

The derivative is: $\frac{dy}{dx} = \frac{2}{7} e^{\frac{2}{7}x}.$

Let me know if you have any questions about the steps! Below are five related questions that may interest you:

1. How do you apply the chain rule to other exponential functions?
2. What are the properties of the exponential function $e^x$?
3. How do you compute higher-order derivatives of exponential functions?
4. How does the chain rule apply to logarithmic functions?
5. What happens to the derivative if the base is not $e$ (e.g., $a^x$)?

Tip: Always rewrite your function carefully when using the chain rule to avoid mixing variables.