To solve the problem of finding the equation of the tangent line to the curve at the specified point $(2,1)$, where the function is given by:

$y = \frac{2x}{x + 2}$

Steps to solve:

1. Find the derivative: The derivative of $y$ with respect to $x$ gives us the slope of the tangent line at any point on the curve. We will use the quotient rule for differentiation since this is a rational function.

   The quotient rule states:

   $\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}$

   Here, $u(x) = 2x$ and $v(x) = x + 2$.

2. Apply the quotient rule:

   • $u'(x) = 2$ and $v'(x) = 1$, $y' = \frac{(2)(x+2) - (2x)(1)}{(x+2)^2}$ Simplify: $y' = \frac{2x + 4 - 2x}{(x + 2)^2} = \frac{4}{(x + 2)^2}$

3. Evaluate the derivative at the point $x = 2$: This gives the slope of the tangent line at $x = 2$.

   $y'(2) = \frac{4}{(2 + 2)^2} = \frac{4}{16} = \frac{1}{4}$

   So, the slope $m$ of the tangent line at $(2,1)$ is $\frac{1}{4}$.

4. Use the point-slope form of the equation of a line: The point-slope form is given by:

   $y - y_1 = m(x - x_1)$

   Substituting $m = \frac{1}{4}$, $x_1 = 2$, and $y_1 = 1$:

   $y - 1 = \frac{1}{4}(x - 2)$

5. Simplify the equation:

   $y - 1 = \frac{1}{4}x - \frac{1}{2}$ $y = \frac{1}{4}x + \frac{1}{2}$

Thus, the equation of the tangent line is:

$y = \frac{1}{4}x + \frac{1}{2}$

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. How do you apply the quotient rule in other examples of rational functions?
2. What are the differences between the quotient rule and product rule in calculus?
3. How do you find the tangent line when the function is a polynomial instead of a rational function?
4. Can you explain how the point-slope form works for finding tangent lines?
5. How would the process change if the curve was implicit (like $x^2 + y^2 = r^2$)?

Tip: When differentiating a quotient, always simplify both the numerator and denominator after applying the quotient rule to avoid calculation errors.