Differentiate the following w.r.t.x. `(x)/((a^(2) + x^(2)))`

Answer

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

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

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

### Step 1: Identify \( u \) and \( v \)
Let:
- \( u = x \)
- \( v = a^2 + x^2 \)

### Step 2: Differentiate \( u \) and \( v \)
Now we find the derivatives:
- \( \frac{du}{dx} = 1 \)
- \( \frac{dv}{dx} = 0 + 2x = 2x \)

### Step 3: Apply the Quotient Rule
Now we can apply the quotient rule:

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

### Step 4: Simplify the Expression
Now we simplify the numerator:

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

This simplifies to:

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

### Final Result
Thus, the derivative of \( y = \frac{x}{a^2 + x^2} \) with respect to \( x \) is:

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

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