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We need to find the equation of the tangent line to the curve \( y = \frac{-3e^x}{x} \) at the point \( (1, -3e) \).

### Steps:
1. **Find the derivative**: To determine the slope of the tangent line, we need to differentiate \( y = \frac{-3e^x}{x} \) with respect to \( x \).
   - Use the quotient rule: \( \frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2} \), where:
     - \( f(x) = -3e^x \) and \( g(x) = x \),
     - \( f'(x) = -3e^x \) and \( g'(x) = 1 \).

2. **Evaluate the derivative at \( x = 1 \)**: This will give the slope \( m \) of the tangent line at the point \( (1, -3e) \).

3. **Find the equation of the tangent line**: Once we have the slope \( m \), we use the point-slope form of a line: \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) = (1, -3e) \).

Let me compute the derivative and the equation of the tangent line next.The derivative of the function \( y = \frac{-3e^x}{x} \) at \( x = 1 \) gives a slope of 0. This means that the tangent line at the point \( (1, -3e) \) is horizontal.

Thus, the equation of the tangent line is simply:

\[
y = -3e
\]

This is a constant line passing through \( (1, -3e) \).

Would you like further details or have any questions? Here are some related problems you might want to explore:

1. How do we apply the quotient rule in other scenarios?
2. What happens if the function involves higher-order derivatives?
3. Can we find the tangent line for non-differentiable points?
4. How does the equation of the normal line relate to the tangent line?
5. Can we approximate the function using Taylor expansion around \( x = 1 \)?

**Tip**: When the slope of a tangent line is zero, the line is horizontal, indicating that the function has a local extremum at that point (either a maximum or minimum).

Find an equation of the tangent line to the given curve at the specified point: y = (-3e^x)/x at (1, -3e).

Solve the problem of finding the equation of the tangent line to the curve y = (-3e^x)/x at the point (1, -3e) using the quotient rule and differentiation. Learn key calculus concepts such as derivatives and tangent lines. Suitable for Grades 11-12 or college-level calculus.

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