Question

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

Answer 1

`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)`