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

1 Answer
Aug 30, 2015

#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#