If `y=(x^(2))/((x+1))"then"(dy)/(dx)"is":-`

Options

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

Answer

To find the derivative of the function \( y = \frac{x^2}{x + 1} \), we will use the quotient rule for differentiation. The quotient rule states that if you have a function in the form \( y = \frac{u}{v} \), then the derivative \( \frac{dy}{dx} \) is given by:

\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

where \( u = x^2 \) and \( v = x + 1 \).

### Step 1: Identify \( u \) and \( v \)

Let:
- \( u = x^2 \)
- \( v = x + 1 \)

### Step 2: Find \( \frac{du}{dx} \) and \( \frac{dv}{dx} \)

Now, differentiate \( u \) and \( v \):
- \( \frac{du}{dx} = 2x \)
- \( \frac{dv}{dx} = 1 \)

### Step 3: Apply the Quotient Rule

Now, substitute \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \) into the quotient rule formula:

\[ \frac{dy}{dx} = \frac{(x + 1)(2x) - (x^2)(1)}{(x + 1)^2} \]

### Step 4: Simplify the Expression

Now, simplify the numerator:

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

Combine like terms:

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

### Step 5: Factor the Numerator

Now, factor out \( x \) from the numerator:

\[ = \frac{x(x + 2)}{(x + 1)^2} \]

### Final Answer

Thus, the derivative \( \frac{dy}{dx} \) is:

\[ \frac{dy}{dx} = \frac{x(x + 2)}{(x + 1)^2} \]

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