Question

What is the derivative of `e^x/(1+e^y)`?

Answer 1

`d/dx(e^x/(1+e^y))= (e^x(1+e^y-e^ydy/dx))/(1+e^y)^2`

`d/dt(e^x/(1+e^y))= (e^x(dx/dt+dx/dt e^y-e^ydy/dt))/(1+e^y)^2`

Explanation:

Assuming that we are differentiation with respect to `x` and that `y` is some function of `x`, we can differentiate implicitly using the quotient rule:

`d/dx(e^x/(1+e^y)) = (e^x(1+e^y)-e^x(e^y dy/dx))/(1+e^y)^2`

If we are differentiating with respect to some `t` and assuming that `x` and `y` are functions of `t`, we get:

`d/dt(e^x/(1+e^y)) = (e^x dx/dt (1+e^y)-e^x(e^y dy/dt))/(1+e^y)^2`

`= (e^x[dx/dt(1+e^y)-e^x(e^y dy/dt)])/(1+e^y)^2`

`= (e^x(dx/dt+dx/dt e^y-e^ydy/dt))/(1+e^y)^2`