Differentiate the following w.r.t. x : y=x^2+3 x^2-5 - Vidyadip

Question

Differentiate the following w.r.t. x : $y=\frac{x^2+3}{x^2-5}$

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Answer

$y=\frac{x^2+3}{x^2-5}$
Differentiating w.r.t. $x$, we get
$\begin{aligned}
\frac{ d y}{ d x} & =\frac{ d }{ d x}\left(\frac{x^2+3}{x^2-5}\right) \\
& =\frac{\left(x^2-5\right) \frac{ d }{ d x}\left(x^2+3\right)-\left(x^2+3\right) \frac{ d }{ d x}\left(x^2-5\right)}{\left(x^2-5\right)^2} \\
& =\frac{\left(x^2-5\right)(2 x)-\left(x^2+3\right)(2 x)}{\left(x^2-5\right)^2} \\
& =\frac{2 x\left(x^2-5-x^2-3\right)}{\left(x^2-5\right)^2} \\
& =\frac{2 x(-8)}{\left(x^2-5\right)^2}=\frac{-16 x}{\left(x^2-5\right)^2}
\end{aligned}$

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