Linearity of differential of vector-valued function In Baby Rudin, Rudin states:

Definition 9.11: Suppose $E$ is an open set in $\mathbb{R}^n$ $f$ maps $E$ into $\mathbb{R}^m$, and $x\in E$. If there exists a linear transformation $A$ of $\mathbb{R}^n$ into $\mathbb{R}^m$ such that $$ \lim_{h\to 0} \frac{\lvert f(x+h) - f(x) + Ah\rvert}{\lvert h\rvert} = 0,$$ then we say that $f$ is differentiable at $x$ and we write $f'(x) = A$.

My confusion stems from the last line. The derivative of a function isn't necessarily
linear (say, for example, $g(x) = x^3$ and $g'(x) =3x^2$). So, when Rudin says $f'(x) = A$, I take it that he doesn't mean the derivative of $f$ is linear. What exactly does he mean then?
I know that $A$ is called the differential so $f'(x) = A$ is the differential of $f$ at $x$, I suppose. Is the differential distinct from the derivative then, since the former is, by requirement, linear while the latter need not be linear?
 A: Your confusion arises because you think of the derivative as a function which matches to each $x\in \mathbb{R}$ the slope of its tangent $f'(x)$. You are right that this function does not need to be linear; for instance, if $f(x)=x^3$, then $f'(x)=3x^2$.
However, consider a fixed point $x_0\in\mathbb{R}$. The derivative which in calculus we think of as a number, can also be thought of as a function which associates to every $x$-increment the $y$-increment of its tangent at $x_0$. For instance, for $f(x)=x^3$ at $x_0=1$ the derivative is $2$; the associated function is $y=2x$. For another example, at $x_0=5$ the derivative is $75$; the associated function is $y=75x$. The meaning of these functions is that to a certain approximation, $f(x)$ behaves roughly like the linear function $y=75x$ around $x_0=5$, and so on. Note that all these functions $y=2x, y=75x$, etc, are linear!
So in calculus we usually think of the derivative at $x_0$ as a number $f'(x_0)$ (the slope of the tangent), and not as this function $y=f'(x_0)x$. The function $y=f'(x_0)x$ is instead called the differential.
In multivariate calculus the same thing is going on. The differential is the linear map that in a sense best approximates the function near a given point. So the multivariate differential generalizes the single-variable differential. (We may still speak of the matrix which represents this linear map; and then this matrix, called the Jacobian, is the generalization of the usual single variable derivative.)
The text you're reading is using the notation $f'(x_0)$ to denote the linear map $x_0 \mapsto Ax_0$, so in that sense they are using $f'(x_0)$ as a generalization of the single-variable differential, rather than the single-variable derivative.
It is the differential which is a linear map, and this is true simply by its definition: it is multiplication by a matrix. This is the function which is the best linear approximation for the original function, at a single given point. The function which outputs the different Jacobian matrices for every varying input point is definitely not necessarily linear.
