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

1 Answer
Steve M
Nov 23, 2016

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

Explanation:

# 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))#

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