Question

if y is a function of x and x=e^t/(e^t+1) show that dy/dt=x(1-x)dy/dx


help me plz

Answer 1

y = f(e^t/(e^t+1))

dy/dt = dy/dx * dx/dt chain rule

= dy/dx * d /dt [ e^t/(e^t+1) ]

= dy/dx [ (e^t+1)e^t - e^2t ]/(e^t+1)^2

now what is x(1-x) ?

[e^t/(e^t+1)][1-e^t/(e^t+1)]
= [ e^t(e^t+1)- e^2t]/(e^t+1)^2
The same !!!

Answer 2

x=e^t/(e^t+1)

dx/dt= -[e^t*e^t)/(e^t+1)^2 + e^t/(e^t+1)
= e^t/(e^t+1) *(-e^t/(e^t+1)+1)
= x * (1-x)

so
dy/dt=dx/dt*dy/dx
and you have it.