I guess I have several definitions of [itex]df[/itex] flying at me, and I am having trouble getting a continuous definition. So in basic Calculus, we are taught [itex]df = f'(x)dx[/itex], and it's taught as sort of a linear approximation of the change of f for small values dx, whch makes sense with the definition of the derivative being a linearization of the change in a function.(adsbygoogle = window.adsbygoogle || []).push({});

[itex]df = f(x+h)-f(x)[/itex]

[itex]f(x+h)-f(x)≈f'(x)h[/itex]

That also makes sense with the higher level definition of a differential being a mapping to the tangent space. I have trouble when I consider a Taylor series based derivation for change in f, it seems to be paradoxical.

[itex]\displaystyle{f(x) = f(a)+f'(a)(x-a)+\frac{f''(a)(x-a)^{2}}{2!}+\frac{f'''(a)(x-a)^{3}}{3!}...}[/itex]

sub [itex]\Delta x = x-a[/itex] and [itex]x = a + \Delta x [/itex]

[itex]\displaystyle{f(\Delta x + a) = f(a)+f'(a)\Delta x+\frac{f''(a)(\Delta x)^{2}}{2!}+\frac{f'''(a)(\Delta x)^{3}}{3!}...}[/itex]

rearrange and you can see the confusion...

[itex]\displaystyle{f(\Delta x + a) -f(a) = f'(a)\Delta x+\frac{f''(a)(\Delta x)^{2}}{2!}+\frac{f'''(a)(\Delta x)^{3}}{3!}...}[/itex]

So now I have different definition of df?? Can anyone explain this to me?

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# Question about the definition of df

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