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

Question

10387 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-03-18T17:27:36

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

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,

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

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

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