Question

Find the derivative of $\frac{x}{1+\tan x}.$

Answer 1

Let $f\left(x\right)=\frac{x}{1+\tan x}$
Differentiating with respect to $x$
$\Rightarrow f^{\prime }\left(x\right)=\frac{d}{dx}\left(\frac{x}{1+\tan x}\right)$
$\Rightarrow f^{\prime }\left(x\right)=\frac{(1+\tan x)\frac{d}{dx}\left(x\right)-\left(x\right)\frac{d}{dx}(1+\tan x)}{(1+\tan x{)}^2}$
$\Rightarrow f^{\prime }\left(x\right)=\frac{1\cdot (1+\tan x)-{\sec }^{2}x\cdot \left(x\right)}{(1+\tan x{)}^2}$
$\therefore f^{\prime }\left(x\right)=\frac{1+\tan x-x{\sec }^{2}x}{(1+\tan x{)}^2}$