Question

Find the derivative of the following functions (it is to be understood that $a,b,c,d,p,q,r$ and $s$ are fixed non-zero constants and $m$ and $n$ are fixed integers) : $(x+\sec x)(x-\tan x)$

Answer 1

Let $f\left(x\right)=(x+\sec x)(x-\tan x)$
Thus by product rule
$f^{\prime }\left(x\right)=(x+\sec x)\frac{d}{dx}(x-\tan x)+(x-\tan x)\frac{d}{dx}(x+\sec x)$
$=(x+\sec x)[\frac{d}{dx}\left(x\right)-\frac{d}{dx}\tan x]+(x-\tan x)[\frac{d}{dx}\left(x\right)+\frac{d}{dx}\sec x]$
$=(x+\sec x)[1-\frac{d}{dx}\tan x]+(x-\tan x)[1+\frac{d}{dx}\sec x]$
$=(x+\sec x)(1-{\sec }^{2}x)+(x-\tan x)(1+\sec x\tan x)$