Question

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

Answer 1

`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))`.