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The Wikipedia article on the Kelly criterion might say more, but here is a simple derivation.
Assuming you have a coin with probability $p$ of coming up heads and odds of $b:1$, the Kelly criterion states:
$$ f^* = \frac{bpq}{b} $$
Where $f^*$ is the fraction of your money pot the Kelly criterion tells you to bet and $q=1p$.
That is:
$$ f^* = \frac{bp(1p)}{b}
= \frac{p(b+1)1}{b} $$
Assuming the strategy is to bet a fraction of your bank roll every round, with $W_0$ as the initial bank roll, $n$ time units and $W_n$ as your winnings at time $n$:
$$ W_n = (1 + br)^{pn} (1  r)^{(1p)n} W_0 $$
Taking logarithms, setting the derivative with respect to $r$ and solving:
$$ \begin{array} . & \frac{d}{dr} \ln(W_n) &= \frac{d}{dr} ( pn \ln(1+br) + (1p)n \ln(1r) + \ln(W_0) ) \\ \to & 0 &= \frac{pnb}{1+br}  \frac{(1p)n}{1r} \\ \to & \frac{(1p)n}{1r} &= \frac{pnb}{1+br} \\ \to & \frac{1p}{1r} &= \frac{pb}{1+br} \\ \to & (1p) (1+br) &= pb (1r) \\ \to & 1  p + br  pbr &= pb  pbr \\ \to & 1  p + br &= pb \\ \to & r &= \frac{pb + p  1}{b} \\ \to & r &= \frac{b(p + 1)  1}{b} \\ \end{array} $$