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To solve the problem and find the equation of the tangent line to the curve \( y = f(x) = \frac{e^x}{x} \) at \( x = 1 \), follow these steps:

### Step 1: Evaluate \( f(1) \)
The equation of the tangent line requires a point on the curve. Let's calculate \( f(1) \):

\[
f(1) = \frac{e^1}{1} = e
\]

So, the point on the curve is \( (1, e) \).

### Step 2: Find the derivative \( f'(x) \)
The slope of the tangent line is given by the derivative \( f'(x) \) evaluated at \( x = 1 \). Use the quotient rule to differentiate \( f(x) = \frac{e^x}{x} \).

The quotient rule states:
\[
\left( \frac{u(x)}{v(x)} \right)' = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}
\]
where \( u(x) = e^x \) and \( v(x) = x \).

- \( u'(x) = e^x \)
- \( v'(x) = 1 \)

Now apply the quotient rule:
\[
f'(x) = \frac{e^x \cdot x - e^x \cdot 1}{x^2} = \frac{e^x(x - 1)}{x^2}
\]

### Step 3: Evaluate \( f'(1) \)
Substitute \( x = 1 \) into the derivative:

\[
f'(1) = \frac{e^1(1 - 1)}{1^2} = \frac{e(0)}{1} = 0
\]

So, the slope of the tangent line at \( x = 1 \) is 0.

### Step 4: Equation of the tangent line
Since the slope of the tangent line is 0 and it passes through the point \( (1, e) \), the equation of the tangent line is horizontal, i.e., \( y = e \).

### Final Answer:
The equation of the tangent line is \( y = e \), which corresponds to option **D**.

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Would you like more details or have any further questions?

### Related Questions:
1. How is the quotient rule applied to differentiate functions?
2. What are the conditions for a tangent line to have a slope of zero?
3. Can you explain the general form of a tangent line equation?
4. How would the tangent line change if we considered another point on the curve?
5. What does the behavior of \( \frac{e^x}{x} \) suggest about its asymptotes?

### Tip:
The quotient rule is essential for differentiating functions of the form \( \frac{f(x)}{g(x)} \). Always ensure that both \( f(x) \) and \( g(x) \) are properly differentiated before simplifying.

Find the equation of the tangent line to the curve y = f(x) = e^x / x for x = 1.

Learn how to find the tangent line to the curve y = e^x / x at x = 1. This calculus problem covers differentiation using the quotient rule and finding the equation of the tangent line. Suitable for advanced high school students and college-level calculus.

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