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,

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

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