# How do you find the derivative of #arctan(e^x)#?

##### 2 Answers

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

#### Explanation:

When tackling the derivative of inverse trig functions. I prefer to rearrange and use Implicit differentiation as I always get the inverse derivatives muddled up, and this way I do not need to remember the inverse derivatives. If you can remember the inverse derivatives then you can use the chain rule.

#y=arctan(e^x) <=> tany=e^x #

Differentiate Implicitly:

# sec^2ydy/dx = e^x # ..... [1]

Using the

# tan^2y+1 -= sec^2y #

# :. (e^x)^2+1=sec^2y #

# :. e^(2x)+1=sec^2y #

Substituting into [1]

# :. (e^(2x)+1)dy/dx=e^x #

# :. dy/dx = (e^x)/(e^(2x)+1) #

#### Explanation:

Using implicit differentiation together with the known derivatives

let

If we draw a right triangle with an angle

#=e^x/sec^2(y)#

#=e^x/(e^(2x)+1)#