Question

Whats the derivative of `ln((e^x)/(1+e^x))`?

Answer 1

`1/(1+e^x)`

Explanation:

`y = ln((e^x)/(1+e^x)) = ln (u), \qquad u = (e^x)/(1+e^x)`

and we use the chain rule `dy/dx = dy/(du) * (du)/dx`

`dy/(du) = 1/u = (1+ e^x)/(e^x)`

from quotient rule:

`(du)/(dx) = (e^x (1+ e^x) - e^x * e^x)/(1+e^x)^2`
`= (e^x )/(1+e^x)^2`

`\implies (dy)/(dx) = (1+ e^x)/(e^x) * (e^x )/(1+e^x)^2 = 1/(1+e^x)`