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.