Obtain the differential coeffcient of the following :
`(x^(2))/(x^(3)+1)`

Answer

To find the differential coefficient of the function \( y = \frac{x^2}{x^3 + 1} \), we will use the quotient rule of differentiation. The quotient rule states that if you have a function in the form \( \frac{u}{v} \), where \( u \) and \( v \) are functions of \( x \), then the derivative is given by:

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

Here, we identify:
- \( u = x^2 \)
- \( v = x^3 + 1 \)

### Step 1: Differentiate \( u \) and \( v \)
First, we need to find the derivatives of \( u \) and \( v \):
- \( \frac{du}{dx} = \frac{d}{dx}(x^2) = 2x \)
- \( \frac{dv}{dx} = \frac{d}{dx}(x^3 + 1) = 3x^2 \)

### Step 2: Apply the Quotient Rule
Now we can apply the quotient rule:
\[ \frac{dy}{dx} = \frac{(x^3 + 1)(2x) - (x^2)(3x^2)}{(x^3 + 1)^2} \]

### Step 3: Simplify the Expression
Now we simplify the numerator:
\[ = \frac{2x(x^3 + 1) - 3x^4}{(x^3 + 1)^2} \]

Expanding the numerator:
\[ = \frac{2x^4 + 2x - 3x^4}{(x^3 + 1)^2} \]
\[ = \frac{-x^4 + 2x}{(x^3 + 1)^2} \]

### Step 4: Factor the Numerator
We can factor out \( x \) from the numerator:
\[ = \frac{x(-x^3 + 2)}{(x^3 + 1)^2} \]

### Final Result
Thus, the differential coefficient of the function \( y = \frac{x^2}{x^3 + 1} \) is:
\[ \frac{dy}{dx} = \frac{x(2 - x^3)}{(x^3 + 1)^2} \]