Question

Find the derivative of $(x+\sec x)(x-\tan x).$

Answer 1

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