I have some questions about the definitions of the derivatives of real valued functions which are mostly to make sure I got things correctly. Are the following statements correct?
The analogous of partial derivatives of functions $f:\Bbb R^n \to \Bbb R$ for functions $g:\Bbb R \to \Bbb R$ is derivatives, correct? The analogous of the derivative of functions such as $f$ for functions such as $g$ can also be defined and it is something different than the usual derivative of $g$, correct?
(If $g(x)=(x-2)^3+3$ then $g'(x)=3(x-2)^2$ which is not a linear map but $Dg=$??)
If the above are correct then we call these linear mappings derivatives because they simply have very similar properties to the derivatives of functions defined in $\Bbb R$.
I hope an answer alleviates this confusion. Thanks in advance