Find the derivative of y w.r.t. x in each of the following :
`y(y+1)=x(x+1)(x+2)`

Answer

To find the derivative of \( y \) with respect to \( x \) in the equation \( y(y + 1) = x(x + 1)(x + 2) \), we will follow these steps:

### Step 1: Expand both sides of the equation
First, we will expand the left-hand side and the right-hand side of the equation.

**Left-hand side:**
\[ y(y + 1) = y^2 + y \]

**Right-hand side:**
\[ x(x + 1)(x + 2) = x(x^2 + 3x + 2) = x^3 + 3x^2 + 2x \]

So, we rewrite the equation as:
\[ y^2 + y = x^3 + 3x^2 + 2x \]

### Step 2: Differentiate both sides with respect to \( x \)
Now, we differentiate both sides of the equation with respect to \( x \).

**Left-hand side:**
Using the product and chain rule:
\[ \frac{d}{dx}(y^2) + \frac{d}{dx}(y) = 2y \frac{dy}{dx} + \frac{dy}{dx} = (2y + 1) \frac{dy}{dx} \]

**Right-hand side:**
Using the power rule:
\[ \frac{d}{dx}(x^3 + 3x^2 + 2x) = 3x^2 + 6x + 2 \]

### Step 3: Set the derivatives equal to each other
Now we equate the derivatives from both sides:
\[ (2y + 1) \frac{dy}{dx} = 3x^2 + 6x + 2 \]

### Step 4: Solve for \( \frac{dy}{dx} \)
To isolate \( \frac{dy}{dx} \), we divide both sides by \( (2y + 1) \):
\[ \frac{dy}{dx} = \frac{3x^2 + 6x + 2}{2y + 1} \]

### Final Answer
Thus, the derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{3x^2 + 6x + 2}{2y + 1} \]

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