To differentiate the function $y = e^{1/x} - \ln(x)$, we'll use the chain rule and the power rule for differentiation.

Step 1: Differentiate $e^{1/x}$

Let $u = \frac{1}{x}$. Then $y_1 = e^u$.

The derivative of $y_1 = e^u$ with respect to $x$ is:

$\frac{dy_1}{dx} = e^u \cdot \frac{du}{dx}$

Now, differentiate $u = \frac{1}{x}$:

$\frac{du}{dx} = -\frac{1}{x^2}$

So,

$\frac{dy_1}{dx} = e^{1/x} \cdot \left(-\frac{1}{x^2}\right) = -\frac{e^{1/x}}{x^2}$

Step 2: Differentiate $-\ln(x)$

The derivative of $y_2 = -\ln(x)$ with respect to $x$ is:

$\frac{dy_2}{dx} = -\frac{1}{x}$

Step 3: Combine the results

Now, combine the derivatives of the two parts:

$\frac{dy}{dx} = -\frac{e^{1/x}}{x^2} - \frac{1}{x}$

So, the derivative of the function $y = e^{1/x} - \ln(x)$ is:

$\boxed{\frac{dy}{dx} = -\frac{e^{1/x}}{x^2} - \frac{1}{x}}$

Would you like a detailed explanation of any part of this solution?

Here are 5 related questions:

1. How do you apply the chain rule in differentiation?
2. What is the derivative of $e^u$ where $u$ is a function of $x$?
3. Can you differentiate $e^{-x}$ using the same method?
4. How do you differentiate natural logarithms?
5. What are some common mistakes when differentiating functions involving exponents and logarithms?

Tip: When differentiating composite functions like $e^{f(x)}$, always remember to apply the chain rule, which requires differentiating both the outer function and the inner function.