# Is #f(x)=(1-e^(2x))/(2x-4)# increasing or decreasing at #x=-1#?

##### 1 Answer

Jan 5, 2016

Increasing.

#### Explanation:

Find the sign of the derivative at

To find

#f'(x)=((2x-4)d/dx(1-e^(2x))-(1-e^(2x))d/dx(2x-4))/(2x-4)^2#

The derivative of each part:

The first requires the chain rule:

#d/dx(1-e^(2x))=-2e^(2x)#

#d/dx(2x-4)=2#

Thus,

#f'(x)=((2x-4)(-2e^(2x))-2(1-e^(2x)))/(2x-4)^2#

#f'(x)=(-4xe^(2x)+10e^(2x)+2)/(2x-4)^2#

Find

#f'(-1)=(-4(-2)e^-2+10e^-2+2)/(-2-4)^2=(18/e^2+2)/36#

Since this is

graph{(1-e^(2x))/(2x-4) [-5.93, 6.56, -2.596, 3.647]}