Question

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

Answer 1

`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))`