What's the derivative of #arctan(e^x)#?

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Answer 1 · 2016-12-01T00:05:20

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

Let # y = arctan(e^x) iff tany=e^x #

Differentiate Implicitly wrt #x#:

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

Using the #tan"/"sec# trig identity:

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

Substituting this result into [1] we get:

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

Hence,

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