1. ## showing

hey pliz help me show that the differentiation of $\displaystyle y=secx-tanx/(secx+tanx) is -2secx/(secx+tanx)^2$

2. Multiply top and bottom by sec(x) +tan(x) to obtain

1/[sec(x)+tan(x)]^2

f ' = -2 (sec(x)+tan(x))^(-3)* (sec(x)tan(x)+sec^2(x))

now just clean it up

3. Hello, Rose!

Show that the derivative of $\displaystyle y\:=\:\frac{\sec x-\tan x}{\sec x+\tan x}$ .is: .$\displaystyle \frac{dy}{dx} \:=\: \frac{-2\sec x}{(\sec x+\tan x)^2}$
We can differentiate this head-on . . .

Quotient Rule:

. . $\displaystyle \frac{dy}{dx} \;=\;\frac{(\sec x + \tan x)(\sec x\tan x - \sec^2\!x) - (\sec x -\tan x)(\sec x\tan x + \sec^2\!x)}{(\sec x +\tan x)^2}$

. . . . $\displaystyle = \;\frac{-\sec x(\sec x + \tan x)(\sec x - \tan x) - \sec x(\sec x - \tan x)(\sec x + \tan x)}{(\sec x +\tan x)^2}$

. . . . $\displaystyle = \;\frac{-2\sec x(\sec x + \tan x)(\sec x -\tan x)}{(\sec x + \tan )^2}$

. . We have: .$\displaystyle \frac{dy}{dx} \;=\;\frac{-2\sec x(\sec x - \tan x)} {\sec x+ \tan x}$

$\displaystyle \text{Multiply by }\frac{\sec x + \tan x}{\sec x + \tan x}\!:$
. . . $\displaystyle \frac{dy}{dx} \;=\;\frac{-2\sec x(\sec x - \tan x)}{\sec x + \tan x}\cdot{\color{blue}\frac{\sec x + \tan x}{\sec x + \tan x}} \;=\; \frac{-2\sec x\overbrace{(\sec^2\!x - \tan^2\!x)}^{\text{This is 1}}}{(\sec x + \tan x)^2}$

Therefore: .$\displaystyle \frac{dy}{dx} \;=\;\frac{-2\sec x}{(\sec x + \tan x)^2}$