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The given problem asks to find the value of \( y' \) when \( x = 0 \), for the equation:

\[
xy^7 + e^{y/3} = e
\]

### Step-by-Step Solution:

1. **Differentiate both sides implicitly with respect to \( x \):**

   \[
   \frac{d}{dx}\left( xy^7 + e^{y/3} \right) = \frac{d}{dx}(e)
   \]

   The right-hand side is a constant, so its derivative is zero:

   \[
   0 = 0
   \]

2. **Differentiate the left-hand side:**

   For the term \( xy^7 \), we apply the product rule:

   \[
   \frac{d}{dx}(xy^7) = \frac{d}{dx}(x) \cdot y^7 + x \cdot \frac{d}{dx}(y^7)
   \]

   This gives:

   \[
   y^7 + x \cdot 7y^6 \cdot y' 
   \]

   For the term \( e^{y/3} \), we use the chain rule:

   \[
   \frac{d}{dx}(e^{y/3}) = e^{y/3} \cdot \frac{1}{3} \cdot y'
   \]

   So the full derivative of the left-hand side is:

   \[
   y^7 + x \cdot 7y^6 y' + \frac{1}{3} e^{y/3} y'
   \]

3. **Combine the full equation:**

   \[
   y^7 + x \cdot 7y^6 y' + \frac{1}{3} e^{y/3} y' = 0
   \]

4. **Substitute \( x = 0 \):**

   \[
   y^7 + \frac{1}{3} e^{y/3} y' = 0
   \]

5. **Solve for \( y' \):**

   We need to solve for \( y' \) at \( x = 0 \). To do so, we first recognize that \( e^{y/3} \) can be simplified by substituting the original equation \( xy^7 + e^{y/3} = e \) at \( x = 0 \), which gives:

   \[
   e^{y/3} = e
   \]

   This simplifies to:

   \[
   y/3 = 1 \quad \Rightarrow \quad y = 3
   \]

   Now substitute \( y = 3 \) into the equation:

   \[
   3^7 + \frac{1}{3} e^{1} y' = 0
   \]

   Simplifying:

   \[
   2187 + \frac{1}{3} e \cdot y' = 0
   \]

   \[
   \frac{1}{3} e y' = -2187
   \]

6. **Final answer for \( y' \):**

   \[
   y' = -\frac{2187 \cdot 3}{e} = -\frac{6561}{e}
   \]

Thus, the value of \( y' \) when \( x = 0 \) is \( -\frac{6561}{e} \).

---

Do you want further details or have any questions?

### Related questions:
1. How do you apply implicit differentiation in other equations?
2. Can this method be extended to second-order derivatives?
3. How does the chain rule work in more complex cases like this?
4. What is the product rule and when should it be applied?
5. How do constants like \( e \) simplify equations in differentiation?

**Tip:** Always check for constants and their derivatives early to simplify calculations efficiently.

Find the value of y' when x = 0 if xy^7 + e^(y/3) = e

This problem uses implicit differentiation, the product rule, and the chain rule to find the value of y' when x = 0 in the equation xy^7 + e^(y/3) = e. Suitable for students studying college-level calculus.

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