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We are asked to compute \(\frac{dy}{dx}\) using the chain rule for the function \( y = \frac{u}{3} + \frac{3}{u} \), where \( u = x - x^7 \), and we must express the result in terms of \( x \) only.

### Step 1: Differentiate \( y \) with respect to \( u \)
First, we differentiate \( y \) with respect to \( u \). Using the standard differentiation rules:

\[
y = \frac{u}{3} + \frac{3}{u}
\]

\[
\frac{dy}{du} = \frac{1}{3} - \frac{3}{u^2}
\]

### Step 2: Differentiate \( u \) with respect to \( x \)
Now, we differentiate \( u = x - x^7 \) with respect to \( x \):

\[
\frac{du}{dx} = 1 - 7x^6
\]

### Step 3: Apply the chain rule
The chain rule tells us that:

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

Substituting the results from the previous steps:

\[
\frac{dy}{dx} = \left( \frac{1}{3} - \frac{3}{u^2} \right) \cdot (1 - 7x^6)
\]

### Step 4: Substitute \( u = x - x^7 \)
Finally, we replace \( u \) with its expression in terms of \( x \) (i.e., \( u = x - x^7 \)):

\[
\frac{dy}{dx} = \left( \frac{1}{3} - \frac{3}{(x - x^7)^2} \right) \cdot (1 - 7x^6)
\]

This is the derivative \(\frac{dy}{dx}\) in terms of \( x \).

Would you like a detailed breakdown of any part of this solution? Here are five related questions to explore further:

1. How does the chain rule work for composite functions?
2. What are some examples of chain rule applications in physics or economics?
3. How do we handle more complex chain rule applications with multiple variables?
4. Can you explain the quotient rule and when it might be used instead of the chain rule?
5. What is the significance of higher-order derivatives, and how do they apply here?

**Tip**: Always simplify expressions as much as possible before applying the chain rule to reduce the risk of algebraic errors.

Compute \( \frac{dy}{dx} \) using the chain rule. State your answer in terms of x only.

\( y = \frac{u}{3} + \frac{3}{u}, \; u = x - x^7 \)

Step-by-step solution for finding dy/dx using the chain rule where y = u/3 + 3/u and u = x - x^7. Learn how to apply the chain rule and simplify expressions in calculus, suitable for advanced high school students.

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