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Let ##F(x)## and ##G(x)## be cumulative distribution functions for independent random variables ##A## and ##B## respectively with probability density functions ##f(x)=F'(x)##, ##g(x)=G'(x)##. Consider the random variable ##C=\max(A,B)##. Let ##H(x)## be the cumulative distribution function of ##C##, with pdf ##h(x)=H'(x)##. Then ##h(x) = f(x)G(x) + F(x)g(x)##. If ##C=x##, it's because either ##A=x## and ##B\leq x## or ##B=x## and ##A\leq x##. But since ##A## and ##B## are independent, to both be smaller than ##x##, the probabilities just multiply. So the cdf is simply ##F(x)G(x)##. Therefore ##\frac{d}{dx} \left(F(x)G(x)\right) = F'(x)G(x) + F(x)G'(x)##.

That's pretty much it! I thought it was kind of neat and wanted to share it.