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