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

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Answer 1 · 2016-06-23T01:22:16

#e^x/sqrt(1-e^(2x))#

There are two methods:

Using the pre-memorized arcsine derivative:

You may already know that the derivative of arcsine is:

#d/dxarcsin(x)=1/sqrt(1-x^2)#

We can apply the chain rule to this for #arcsin(e^x)#:

#d/dxarcsin(e^x)=1/sqrt(1-(e^x)^2)*d/dxe^x#

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

Without knowing the arcsine derivative:

Let

#y=arcsin(e^x)#

Thus:

#sin(y)=e^x#

Differentiate both sides (the chain rule will be used on the left-hand side!):

#y^'*cos(y)=e^x#

#y^'=e^x/cos(y)#

Note that we should express #cos(y)# in terms of #sin(y)#, since we know that #sin(y)=e^x#.

We know that

#sin^2(y)+cos^2(y)=1" "=>" "cos(y)=sqrt(1-sin^2(y))#

#y^'=e^x/sqrt(1-sin^2(y))#

And since #sin(y)=e^x#:

#y^'=e^x/sqrt(1-(e^x)^2)=e^x/sqrt(1-e^(2x))#