Differentiate
$\frac{\left(x^{2}-1\right)}{\left(x^{2}+7 x+1\right)}$
To find: Differentiation of $\frac{\left(x^{2}-1\right)}{\left(x^{2}+7 x+1\right)}$
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 $u=\left(x^{2}-1\right)$ and $v=\left(x^{2}+7 x+1\right)$
$u^{\prime}=\frac{d u}{d x}=\frac{d\left(x^{2}-1\right)}{d x}=2 x$
$v^{\prime}=\frac{d v}{d x}=\frac{d\left(x^{2}+7 x+1\right)}{d x}=2 x+7$
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)
$\left[\frac{\left(x^{2}-1\right)}{\left(x^{2}+7 x+1\right)}\right]=\frac{2 x \times\left(x^{2}+7 x+1\right)-\left(x^{2}-1\right) \times(2 x+7)}{\left(x^{2}+7 x+1\right)^{2}}$
$=\frac{2 x^{3}+14 x^{2}+2 x-2 x^{3}-7 x^{2}+2 x+7}{\left(x^{2}+7 x+1\right)^{2}}$
$=\frac{7 x^{2}+4 x+7}{\left(x^{2}+7 x+1\right)^{2}}$
Ans $)=\frac{7 x^{2}+4 x+7}{\left(x^{2}+7 x+1\right)^{2}}$
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