Is there a name for a function of the form $y(x;p) = p f(x) + (1-p) g(x)$, where $p$ is some scalar between 0 and 1, and $f$ and $g$ are functions?
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It's a convex combination of $f$ and $g$. For more information see http://en.wikipedia.org/wiki/Convex_combination.
If $f$ and $g$ are differentiable, then so is $y$ and the derivative with respect to $x$ is given by $pf'(x)+(1-p)g'(x)$, so the derivative of a convex combination is the convex combination of the individual derivatives.
sranthrop
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Cool, thanks. Is "convex" an overloaded term here? As in, when I think of functions being convex, it seems to have a special meaning. Although my impression of convexity here might be a special case of something more general then? – rhombidodecahedron Feb 23 '15 at 17:11
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Also: is there a known formula for the derivative of a convex combination? – rhombidodecahedron Feb 23 '15 at 17:12
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1I added some detail :) – sranthrop Feb 23 '15 at 17:16
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Thanks for your help! I'll accept when the time limit expires. By the way, do you know anything about the derivative of the log of a convex combination? In reality I have $\log(y(x;p))$. – rhombidodecahedron Feb 23 '15 at 17:23
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Actually, wolframalpha was able to answer that. Thanks again for your help. – rhombidodecahedron Feb 23 '15 at 17:24
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1You are very welcome. – sranthrop Feb 23 '15 at 17:25