Differentiate

To find: Differentiation of $\frac{x}{\left(a^{2}+x^{2}\right)}$

Formula used: (i) $\left(\frac{u}{v}\right)^{\prime}=\frac{u^{\prime} v-u v^{\prime}}{v^{2}}$ where $v \neq 0$ (Quotient rule)

(ii) $\frac{d x^{n}}{d x}=n x^{n-1}$

Let us take $u=(x)$ and $v=\left(a^{2}+x^{2}\right)$

$u^{\prime}=\frac{d u}{d x}=\frac{d(x)}{d x}=1$

$v^{\prime}=\frac{d v}{d x}=\frac{d\left(a^{2}+x^{2}\right)}{d x}=2 x$

Putting the above obtained values in the formula:-

$\left(\frac{u}{v}\right)^{\prime}=\frac{u^{\prime} v-u v^{\prime}}{v^{2}}$ where $v \neq 0$ (Quotient rule)

$\left[\frac{x}{\left(a^{2}+x^{2}\right)}\right]^{\prime}=\frac{1 \times\left(a^{2}+x^{2}\right)-(x) \times(2 x)}{\left(a^{2}+x^{2}\right)^{2}}$

$=\frac{a^{2}+x^{2}-2 x^{2}}{\left(a^{2}+x^{2}\right)^{2}}$

$=\frac{a^{2}-x^{2}}{\left(a^{2}+x^{2}\right)^{2}}$

Ans $)=\frac{a^{2}-x^{2}}{\left(a^{2}+x^{2}\right)^{2}}$