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To find \(\frac{dy}{dx}\) given that \(x = e^y\), we can follow these steps:

1. **Express \(y\) in terms of \(x\)**:
   Since \(x = e^y\), we can take the natural logarithm of both sides:
   \[
   \ln(x) = \ln(e^y) \implies \ln(x) = y
   \]
   So, \(y = \ln(x)\).

2. **Differentiate both sides with respect to \(x\)**:
   \[
   \frac{d}{dx}(y) = \frac{d}{dx}(\ln(x))
   \]
   Using the chain rule on the left-hand side and the known derivative of \(\ln(x)\) on the right-hand side, we get:
   \[
   \frac{dy}{dx} = \frac{1}{x}
   \]

So, \(\frac{dy}{dx} = \frac{1}{x}\).

Let's check the options provided:

(a) \(\frac{1 - \frac{y}{x}}{x}\)

(b) \(1 + \frac{y}{x}\)

(c) \(\frac{x - y}{x \log x}\)

(d) \(\frac{x + y}{x \log x}\)

None of these options match \(\frac{dy}{dx} = \frac{1}{x}\). This suggests that there might be a typographical error in the problem statement or in the provided options, or it might require further simplification. 

To double-check, we will solve this again using implicit differentiation on the original equation \(x = e^y\):

1. Differentiate both sides with respect to \(x\):
   \[
   \frac{d}{dx}(x) = \frac{d}{dx}(e^y)
   \]
   The derivative of \(x\) with respect to \(x\) is 1, and the derivative of \(e^y\) with respect to \(x\) using the chain rule is \(e^y \cdot \frac{dy}{dx}\).

   So, we get:
   \[
   1 = e^y \cdot \frac{dy}{dx}
   \]

2. Solve for \(\frac{dy}{dx}\):
   \[
   \frac{dy}{dx} = \frac{1}{e^y}
   \]
   Since \(x = e^y\), we substitute \(e^y\) with \(x\):
   \[
   \frac{dy}{dx} = \frac{1}{x}
   \]

So, \(\frac{dy}{dx} = \frac{1}{x}\) again.

The correct derivative should indeed be \(\frac{1}{x}\), which does not match any of the given options. Therefore, there might be an error in the provided choices. If this happens often, I would recommend double-checking with a teacher or an authoritative source.

Do you want more details or have any questions?

Here are five more questions for you to consider:
1. How do you find the second derivative of a function implicitly defined?
2. What is the method to solve differential equations of the form \(\frac{dy}{dx} = f(x, y)\)?
3. How do you find the critical points of a function?
4. What is the process for using L'Hôpital's rule for indeterminate forms?
5. How do you compute the integral of \(\frac{1}{x}\)?

**Tip:** When differentiating implicitly, remember to apply the chain rule carefully to account for dependent variables.

Learn how to solve the implicit differentiation problem where x = e^y. Find the derivative dy/dx using the chain rule and natural logarithm differentiation. Detailed step-by-step solution provided.

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