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

1 Answer
mason m
Sep 10, 2016

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

Explanation:

Alternative approach (basically the same as the one already presented):

Note that y=arcsin(e^x) can be manipulated to say that sin(y)=e^x.

Take the derivative of both sides with respect to x. Recall that the chain rule will be used on the left side: cos(y)dy/dx=e^x

Solving for dy/dx gives dy/dx=e^x/cos(y). We know that sin(y)=e^x, so we can write cos(y) as cos(y)=sqrt(1-sin^2(y). Note that this relationship comes from the Pythagorean identity sin^2(y)+cos^2(y)=1.

Thus, we have dy/dx=e^x/sqrt(1-sin^2(y)). And, since we know that sin(y)=e^x, this gives a fully simplified derivative of dy/dx=e^x/sqrt(1-e^(2x)).

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