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(Sorry for bad English.)

Let $f:\mathbb R^n\to\mathbb R^m$, $x\in\mathbb R^n$. What is a drivative $f'(x)$? It is the linear map $f'(x):\mathbb R^n\to\mathbb R^m$, $h\mapsto (f'(x))(h)=:f'(x)h$.

But what is a differential $df$? Usually people say $df(x)=f'(x)h$, but what is $h$? I tried to write it more correctly: $df(x,h)=f'(x)h$. I compared it with above. I see that simply $df=f'$. I'm confused a bit.

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Many many posts have been written on related topics. You might want to check the following: one, two, three, four, five, six, seven, eight, nine, ten... – Srivatsan Jan 25 '12 at 16:55
Of course, these posts are about related topics, so I thought they would be helpful to you. Not all of them address your question directly. – Srivatsan Jan 25 '12 at 17:06
Yes, it is entirely reasonable to take $\mathrm{d}f = f'$. We can even define $dx$ to be the constant function whose value is the identity matrix and say $\mathrm{d}f(x) = f'(x) \, \mathrm{d}x$ – voila, chain rule! – Zhen Lin Jan 25 '12 at 18:23

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