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.