`d/dx[(x+1)^3/x]=?`

Options

A. `(3(x+1)^2)/x+(x+1)^3/x^2`
B. `(3(x+1)^2)/x.(x+1)^3/x^2`
C. `(2(x+1)^2)/x-(x+1)^3/x^2`
D. `(3(x+1)^2)/x+(2(x+1)^3)/x^2`

Answer

To solve the problem \( \frac{d}{dx} \left[ \frac{(x+1)^3}{x} \right] \), we will use the quotient rule for differentiation. The quotient rule states that if you have a function in the form \( \frac{u}{v} \), then its derivative is given by:

\[ \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

### Step-by-Step Solution:

1. **Identify \( u \) and \( v \)**:
- Let \( u = (x+1)^3 \)
- Let \( v = x \)

2. **Find \( \frac{du}{dx} \)**:
- To differentiate \( u = (x+1)^3 \), we apply the chain rule:
\[ \frac{du}{dx} = 3(x+1)^2 \cdot \frac{d}{dx}(x+1) = 3(x+1)^2 \cdot 1 = 3(x+1)^2 \]

3. **Find \( \frac{dv}{dx} \)**:
- Since \( v = x \):
\[ \frac{dv}{dx} = 1 \]

4. **Apply the Quotient Rule**:
- Now substitute \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \) into the quotient rule formula:
\[ \frac{d}{dx} \left( \frac{(x+1)^3}{x} \right) = \frac{x \cdot 3(x+1)^2 - (x+1)^3 \cdot 1}{x^2} \]

5. **Simplify the expression**:
- Distributing in the numerator:
\[ = \frac{3x(x+1)^2 - (x+1)^3}{x^2} \]
- Factor out \( (x+1)^2 \):
\[ = \frac{(x+1)^2 (3x - (x+1))}{x^2} \]
- Simplifying \( 3x - (x + 1) \):
\[ = \frac{(x+1)^2 (3x - x - 1)}{x^2} = \frac{(x+1)^2 (2x - 1)}{x^2} \]

6. **Final Result**:
- Thus, the derivative is:
\[ \frac{d}{dx} \left[ \frac{(x+1)^3}{x} \right] = \frac{(x+1)^2 (2x - 1)}{x^2} \]