Are derivatives linear maps? I am reading Rudin and I am very confused what a derivative is now. I used to think a derivative was just the process of taking the limit like this
$$\lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{(x+h)-x}$$
But between Apostol and Rudin, I am confused in what sense total derivatives are derivatives. 
Partial derivatives much more resemble the usual derivatives taught in high school 
$$f(x,y) = xy$$
$$\frac{\partial f}{\partial x} = y$$
But the Jacobian doesn't resemble this at all. And according to my books it is a linear map.  
If derivatives are linear maps, can someone help me see more clearly how my intuitions about simpler derivatives relate to the more complicated forms? I just don't understand where the limits have gone, why its more complex, and why the simpler forms aren't described as linear maps. 

If $f:M\to N$ is some (possibly nonlinear) function (here I have in mind a diffeomorphism), then the Jacobian $J_f$ can be viewed as a linear map taking tangent vectors at some point $p \in M$ and returning a tangent vector at $f(p) \in N$.
Let's consider the case where $M$ and $N$ are both $\mathbb R^3$.  $f$ is some vector field (or perhaps it could be viewed as a transformation of the vector space), and the Jacobian $J_f$ tells us about directional derivatives of $f$.  Given a direction $v$ at a point $p$, $J_f(v) = (v \cdot \nabla) f|_p$.
If $M$ and $N$ are both $\mathbb R^1$, then what do we have?  There's only one linearly independent tangent vector at each point, so $J_f$ is uniquely determined by some unit vector $\hat x$ and we get $J_f(\hat x) = \frac{df}{dx} \hat x$.  The Jacobian here just tells us how a unit length is stretched or shrunk when we view $f$ as a transformation of the real line.
This is the way in which a Jacobian is a linear map:  it tells us how directions in the domain correspond to directions in the range.  And even 1d derivatives can be seen in this way.  The components of the Jacobian are still the partial derivatives you're familiar with.  We're just using those partial derivatives to talk about transformations of directions under some function, some map.
