What is the derivative of #[e^x / (1 - e^x)]#?

1 Answer
Jul 31, 2015

#y^' = e^x/(1- e^x)^2#

Explanation:

The quotient rule will work just fine for this function because you can write it as

#y = e^x/(1-e^x) = f(x)/g(x)#

In such cases, the derivative of the function can be found by

#color(blue)(d/dx(y) = (f^'(x) * g(x) - f(x) * g^'(x))/[g(x)]^2#, with #g(x)!=0#.

Apart from this, all you really need to know is that

#d/dx(e^x) = e^x#

So, the derivative of #y# will be

#d/dx(y) = ([d/dx(e^x)] * (1 - e^x) - e^x * d/dx(1 - e^x))/(1 - e^x)^2#

#y^' = (e^x * (1-e^x) - e^x * (-e^x))/(1-e^x)^2#

#y^' = (e^x - color(red)(cancel(color(black)(e^(2x)))) + color(red)(cancel(color(black)(e^(2x)))))/(1 - e^x)^2#

#y^' = color(green)(e^x/(1- e^x)^2)#