Question #51ad1

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Answer 1 · 2016-07-26T21:25:03

Please see the explanation section below.

If #y# is a function of #x# and #x# a function of #t#, then the chain rule tells us that

#dy/dt = dy/dx dx/dt#.

The problem here therefore amounts to showing that

#dx/dt = x(1-x)#.

With #x=(e^t)/(1+e^t)#, we use the quotient rule to get

#dx/dt = (e^t(1+e^t)-e^t(e^t))/(1+e^t)^2#

# = e^t/(1+e^t)^2#.

Substituting for #x# in #x(1-x)#, we get

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

# = e^t/(1+e^t) ((1+e^t-e^t)/(1+e^t))#

# = e^t/(1+e^t)^2#.

We conclude that #dx/dt = x(1-x)# as required.