# Is #f(x)=1/e^x# increasing or decreasing at #x=0#?

##### 1 Answer

Feb 6, 2016

Decreasing

#### Explanation:

First, recognize that

#f(x)=e^-x#

To determine whether this is increasing or decreasing at a point, we use the sign of the first derivative.

- If
#f'(0)<0# , then#f(x)# is decreasing at#x=0# . - If
#f'(0)>0# , then#f(x)# is increasing at#x=0# .

Now, to find the derivative, we will use the chain rule. In the case of an exponential function with base

#d/dx(e^u)=e^u*u'#

Here,

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

Find the sign of the derivative at

#f'(0)=-e^-0=-e^0=-1#

Recall that anything (other than

Since

We can check a graph of the original function:

graph{e^-x [-10, 15.31, -4.05, 8.6]}