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The tangent line to the curve \( y = \left( \frac{x+1}{x-3} \right)^2 \) at \( x = 4 \) is given by the equation:

\[
y = 185 - 40x
\]

This line has a slope of \(-40\) and passes through the point \((4, 25)\).

Would you like further details or clarifications? Here are some related questions for you to explore:

1. How do you find the derivative of a quotient function?
2. What is the geometric meaning of the slope of a tangent line?
3. Can you find the tangent line to this function at another point?
4. How does the shape of the graph change as \( x \) approaches 3, where the function has a vertical asymptote?
5. How can you verify the point of tangency and slope using a graph?

**Tip:** When solving problems involving tangent lines, it's helpful to visualize the function and its tangent graphically to better understand the solution.

Find the tangent to y = ((x + 1) / (x - 3))^2 at x = 4.

Learn how to find the tangent line to the function y = ((x + 1) / (x - 3))^2 at x = 4. This problem covers key calculus concepts such as derivatives, the quotient rule, and tangent lines. Suitable for high school students studying calculus.

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