Differentiate (x-3) (x^ 2 +4 ) 3 x^ 2 +4 x+5 w.r.t. x. - Vidyadip

Question

Differentiate $\sqrt{\frac{(x-3)\left(x^{2}+4\right)}{3 x^{2}+4 x+5}} w.r.t. x.$

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Answer

Let $y = \sqrt{\frac{(x-3)\left(x^{2}+4\right)}{\left(3 x^{2}+4 x+5\right)}}$
Taking logarithm on both sides, we have
$\log y = \frac{1}{2}[\log (x – 3) + \log (x^2 + 4) – \log (3x^2 + 4x + 5)]$
Now, differentiating both sides $w.r.t. x,$ we get
$\frac{1}{y} \cdot \frac{d y}{d x} = \frac{1}{2}\left[\frac{1}{(x-3)}+\frac{2 x}{x^{2}+4}-\frac{6 x+4}{3 x^{2}+4 x+5}\right]$
or $\frac{d y}{d x} = \frac{y}{2}\left[\frac{1}{(x-3)}+\frac{2 x}{x^{2}+4}-\frac{6 x+4}{3 x^{2}+4 x+5}\right]$
$\Rightarrow ~\frac{dy}{dx} = \frac{1}{2} \sqrt{\frac{(x-3)\left(x^{2}+4\right)}{3 x^{2}+4 x+5}}\left[\frac{1}{(x-3)}+\frac{2 x}{x^{2}+4}-\frac{6 x+4}{3 x^{2}+4 x+5}\right]$

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