https://file.aiquickdraw.com/mathbot/file/1729118359729st6cmgc4.jpg

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.

Write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x. Given y = e^(2/7 * x).

Learn how to find the derivative of the exponential function y = e^(2/7x) by applying the chain rule. The solution includes rewriting the function in the form y = f(u) and u = g(x), then differentiating step-by-step. Suitable for students in Grades 11-12.

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