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.

Understand how to differentiate the function y = e^(1/x) - ln(x) using the chain rule. This detailed guide covers key differentiation concepts, including exponential and logarithmic functions, and is suitable for students in Grades 11-12.

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