How do you find the derivative of $arcsin(e^x)$ ?

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Answer 1 · 2016-11-23T22:56:17

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

$y= arcsin(e^x) <=> siny=e^x$

Differentiating (Implicitly) wrt $x$ :

$cosydy/dx = e^x$

Using the Identity $sin^y+cos^2y -= 1$
$:. (e^x)^2 + cos^2y = 1$
$:. e^(2x) + cos^2y = 1$
$:. cos^2y = 1 -e^(2x)$
$:. cosy = sqrt(1 -e^(2x))$

And so:
$:. sqrt(1 -e^(2x))dy/dx = e^x$
$:. dy/dx = e^x/sqrt(1 -e^(2x))$

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