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We define derivatives of functions as linear transformations of $R^n \to R^m$. Now talking about the derivative of such linear transformation , as we know if $x \in R^n$ , then

$A(x+h)-A(x)=A(h)$, because of linearity of $A$, which implies that $A'(x)=A$ where , $A'$ is derivative of $A$ . What does this mean? I am not getting the point I think.

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    What does "what does this essentially mean" mean?2012-06-04
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    @QiaochuYuan : It means how do i understand it. I didn't understand it .2012-06-04
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    I agree with Qiaochu's objection. Anyway, I'd say: the derivative of a mapping between normed spaces is defined as "the best linear approximation" (in a sense to be quantified precisely). Clearly the best linear approximation of a linear map is the map itself.2012-06-04
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    Perhaps considering the one-dimensional case will give some clarity. Take $f(x) = ax$, which is a linear transformation from $\mathbb R$ to $\mathbb R$. Then $f'(x) = a$.2012-06-04
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    https://math.berkeley.edu/~bmcmilla/solutions/HWsolutions10.pdf2015-03-23
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    I think it is a fair question since this seems counter-intuitive to what one learns in standard introductory calculus in the US. See below for my answer.2016-03-14

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