To differentiate the function \( y = \frac{(x+2)^2}{x} \), we can use the quotient rule, which is given by:

\[
\left(\frac{u}{v}\right)' = \frac{v \cdot u' - u \cdot v'}{v^2}
\]

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

### Step 1: Differentiate \( u = (x+2)^2 \)
Using the chain rule:

\[
u' = 2(x+2) \cdot \frac{d}{dx}(x+2) = 2(x+2) \cdot 1 = 2(x+2)
\]

### Step 2: Differentiate \( v = x \)
\[
v' = 1
\]

### Step 3: Apply the Quotient Rule
Now, apply the quotient rule:

\[
y' = \frac{x \cdot 2(x+2) - (x+2)^2 \cdot 1}{x^2}
\]

### Step 4: Simplify the Expression
First, expand the terms in the numerator:

\[
y' = \frac{2x(x+2) - (x+2)^2}{x^2}
\]

Expand \( 2x(x+2) \):

\[
2x(x+2) = 2x^2 + 4x
\]

Expand \( (x+2)^2 \):

\[
(x+2)^2 = x^2 + 4x + 4
\]

Now substitute these into the expression for \( y' \):

\[
y' = \frac{2x^2 + 4x - (x^2 + 4x + 4)}{x^2}
\]

Simplify the numerator:

\[
y' = \frac{2x^2 + 4x - x^2 - 4x - 4}{x^2}
\]

\[
y' = \frac{x^2 - 4}{x^2}
\]

Finally, you can simplify further:

\[
y' = 1 - \frac{4}{x^2}
\]

So, the derivative of the function is:

\[
y' = 1 - \frac{4}{x^2}
\]

Would you like more details, or do you have any questions?

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Here are 5 related questions to explore further:

1. How do you differentiate functions involving higher powers of polynomials?
2. Can the product rule be used instead of the quotient rule in some cases?
3. How does the chain rule apply when differentiating composite functions?
4. What are the steps to differentiate a function with a variable in both the numerator and denominator?
5. How can you find the second derivative of the function \( y = \frac{(x+2)^2}{x} \)?

**Tip:** When simplifying expressions, always check for common factors in the numerator and denominator before expanding to avoid unnecessary complexity.

Learn how to differentiate the function y = ((x+2)^2) / x using the quotient rule and polynomial expansion. Detailed step-by-step solution with explanations and simplifications. Suitable for advanced high school students and anyone learning calculus.

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