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Continuity and Differentiability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity and Differentiability3 Marks
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 (x2 + 4) – log (3x2 + 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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