Question

What is the derivative of this function `y=csc^-1(e^x)`?

Answer 1

The answer is `=-e^-x/sqrt(1-e^(-2x))`

Explanation:

The function is

`y=arc csc(e^x)`

Therefore,

`cscy=e^x`

Differentiating wrt `x`

`(1/siny)'dy/dx=e^x`

`(-1/sin^2y*cosy)dy/dx=e^x`

`dy/dx=-tany*siny*e^x`

`siny=1/e^x`

`tany=siny/cosy=(e^(-x)/sqrt(1-e^(-2x)))`

Therefore,

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

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