Differentiation — Maths STD 12 Science — Question

Differentiating w.r.t. x, we get

$\begin{aligned} \frac{d y}{d x} & =\frac{d}{d x}\left[3 \log \tan x+4 \log \sin x+7 \log \left(x^2+7\right)\right] \\ & =3 \frac{d}{d x}(\log \tan x)+4 \frac{d}{d x}(\log \sin x)+7 \frac{d}{d x}\left[\log \left(x^2+7\right)\right] \\ & =3 \times \frac{1}{\tan x} \cdot \frac{d}{d x}(\tan x)+4 \times \frac{1}{\sin x} \cdot \frac{d}{d x}(\sin x)+ \\ & =3 \times \frac{1}{\tan x} \cdot \sec 2 x+4 \times \frac{1}{\sin x} \cdot \cos x+7 \times \frac{1}{x^2+7} \cdot(2 x+0) \\ & =3 \times \frac{\cos x}{\sin x} \times \frac{1}{\cos ^2 x}+4 \cot x+\frac{1}{x^2+7} \cdot \frac{d}{d x}\left(x^2+7\right) \\ & =\frac{6}{2 \sin x \cos x}+4 \cot x+\frac{14 x}{x^2+7} \\ & =\frac{6}{\sin 2 x}+4 \cot ^2 x+\frac{14 x}{x^2+7} \\ & =6 \operatorname{cosec} 2 x+4 \cot x+\frac{14 x}{x^2+7}\end{aligned}$