To differentiate the function \( y = \frac{(x+1)^3}{x} \), we will use the **quotient rule** and **chain rule**. The quotient rule states:

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

Here, let:
- \( u(x) = (x+1)^3 \)
- \( v(x) = x \)

### Step 1: Differentiate \( u(x) = (x+1)^3 \)

Using the chain rule:
\[
u'(x) = 3(x+1)^2
\]

### Step 2: Differentiate \( v(x) = x \)

\[
v'(x) = 1
\]

### Step 3: Apply the Quotient Rule

Now, apply the quotient rule:
\[
y'(x) = \frac{x \cdot 3(x+1)^2 - (x+1)^3 \cdot 1}{x^2}
\]

### Step 4: Simplify the Expression

Factor out \( (x+1)^2 \) from the numerator:
\[
y'(x) = \frac{(x+1)^2 [3x - (x+1)]}{x^2}
\]
Simplify inside the brackets:
\[
y'(x) = \frac{(x+1)^2 (2x - 1)}{x^2}
\]

So, the derivative of \( y = \frac{(x+1)^3}{x} \) is:
\[
y'(x) = \frac{(x+1)^2 (2x - 1)}{x^2}
\]

Would you like further details or clarification on any step?

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Here are 5 related questions you might find useful:
1. How does the quotient rule apply to more complex functions?
2. What are the key differences between the product rule and the quotient rule?
3. How does the chain rule apply to composite functions?
4. What are some common mistakes made when using the quotient rule?
5. Can the quotient rule be applied when the denominator is a constant?

**Tip:** When using the quotient rule, always simplify the result step by step to avoid algebraic errors.

This guide explains how to differentiate the function y = (x+1)³/x using both the quotient rule and chain rule. Step-by-step simplifications and key calculus concepts are provided for students at the college/university level.

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