Find an equation of the tangent line to the curve at the given point.
y =
4x
x + 1
,    (3, 3)
y =

To find the equation of the tangent line to the curve at the given point, we need to:

1. Differentiate the given function to find the slope of the tangent line at the point \((3, 3)\).
2. Use the point-slope form of the equation of a line.

Given the function \( y = \frac{4x}{x + 1} \), we can follow these steps:

### Step 1: Differentiate the function \( y = \frac{4x}{x + 1} \)
We will apply the quotient rule for differentiation, which is:
\[
\left( \frac{f(x)}{g(x)} \right)' = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}
\]
Here, \( f(x) = 4x \) and \( g(x) = x + 1 \).

- \( f'(x) = 4 \)
- \( g'(x) = 1 \)

Now, applying the quotient rule:
\[
y' = \frac{4(x + 1) - 4x(1)}{(x + 1)^2}
\]
Simplifying the numerator:
\[
y' = \frac{4(x + 1) - 4x}{(x + 1)^2} = \frac{4x + 4 - 4x}{(x + 1)^2} = \frac{4}{(x + 1)^2}
\]

### Step 2: Find the slope at \( x = 3 \)
Now, substitute \( x = 3 \) into the derivative to find the slope of the tangent line at the point \( (3, 3) \):
\[
y'(3) = \frac{4}{(3 + 1)^2} = \frac{4}{16} = \frac{1}{4}
\]
So, the slope of the tangent line at \( x = 3 \) is \( \frac{1}{4} \).

### Step 3: Use the point-slope form of the line equation
The point-slope form is:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope and \( (x_1, y_1) \) is the given point. Using \( m = \frac{1}{4} \) and \( (x_1, y_1) = (3, 3) \):
\[
y - 3 = \frac{1}{4}(x - 3)
\]
Simplifying:
\[
y - 3 = \frac{1}{4}x - \frac{3}{4}
\]
\[
y = \frac{1}{4}x + \frac{9}{4}
\]

Thus, the equation of the tangent line is:
\[
y = \frac{1}{4}x + \frac{9}{4}
\]

Would you like more details or explanations on any of these steps?

Here are five related questions to explore this topic further:

1. How do you use the product rule to differentiate functions?
2. What is the geometric significance of the derivative at a point on a curve?
3. How do you find tangent lines for parametric equations?
4. What is the relationship between a derivative and a normal line?
5. How do you apply the chain rule in differentiation?

**Tip**: For rational functions, using the quotient rule helps you differentiate efficiently by focusing on both numerator and denominator simultaneously.

Find an equation of the tangent line to the curve at the given point: y = 4x / (x + 1), (3, 3).

Learn how to find the equation of the tangent line to the curve y = 4x / (x + 1) at the point (3, 3). This calculus problem involves using the quotient rule for differentiation and applying the point-slope form. Suitable for students in Grades 11-12.

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