Question

Differentiate: $\sqrt{\frac{(x-3)(x^2+4)}{3x^2+4x+5}}$

Answer 1

$y=\sqrt{\frac{\left(x-3\right)\left(x^2+4\right)}{3x^2+4x+5}}$

Taking $log$ on both sides

$\log y=\frac{1}{2}\log \left[\frac{\left(x-3\right)\left(x^2+4\right)}{3x^2+4x+5}\right]$

$\Rightarrow \log y=\frac{1}{2}\left[\log \left(x-3\right)+\log \left(x^2+4\right)-\log \left(3x^2+4x+5\right)\right]$

Differentiating w.r.t.x

$\frac{1}{y}\frac{dy}{dx}=\frac{1}{2}\left[\frac{1}{x-3}+\frac{1}{x^2+4}\times 2x-\frac{1\left(6x+4\right)}{3x^2+4x+5}\right]$

$\Rightarrow \frac{dy}{dx}=\frac{1}{2}\sqrt{\frac{\left(x-3\right)\left(x^2+4\right)}{3x^2+4x+5}}\left[\frac{1}{x-3}+\frac{2x}{x^2+4}-\frac{6x+4}{3x^2+4x+5}\right]$