If $f(z)$ is a single variable function(continuous and differentiable) with $z \in R ,z>0$ is there a way to expand this differential?? $$df(xy)$$ meaning if $z=x*y$ with $x,y>0$
$*$ is the simple multiplication.
How to expand $df(xy)$
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$\begingroup$
calculus
multivariable-calculus
derivatives
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0What does expand a differential mean? You can consider the map $g(x,y)=f(x\cdot y)$ which is a two variable map, and compute its differential (I am not sure if this is what you meant). – 2017-02-19
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0for example $d(x*y)=xdy+ydx$. – 2017-02-19
1 Answers
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Differentials satisfy the chain rule:
$$ \mathrm{d} f(u) = f'(u)\, \mathrm{d} u $$
and they respect equations, such as $u = xy$:
$$ \mathrm{d} f(xy) = f'(xy) \, \mathrm{d}(xy) $$
so you'd get
$$ \mathrm{d} f(xy) = f'(xy) (x \, \mathrm{d}y + y \, \mathrm{d}x ) $$
And, incidentally, this equation still holds even if $x$ and $y$ are dependent on one another, and even if one (or both!) are constant!