Does the concept of a derivative a rate of change work for n dimensions? I am trying to understand what exactly a derivative is. I understand the total derivative is a linear map. But I don't understand what happens to the idea of a rate. 
In high school calc, one is taught the derivative is a rate of change. For example, 
$$\frac{dx}{dt}$$ 
But for a vector function like $$f(x,y) = xy^2 \mathbf{\hat{i}} + x^5 \mathbf{\hat{j}} + \sqrt{y}\mathbf{\hat{k}}$$
What does rate of change mean if I take the total or partial derivative of this? 
Should I think of a partial and total derivative like a collection of rates? 
 A: No, you shouldn't think of the total derivative as a rate of change. The generalisation of the concept of the derivative follows from another perspective of understanding it.
One can see it is a rate of change from the following: 
$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$ which is the same as $\lim_{\Delta x\to0}\frac{\Delta y}{\Delta x} $.
However, this leads to some deeply related concept, approximating through a linear function. (You'd see this easily if you knew some Taylor series)
$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$
This means that 
$f'(x)h=f(x+h)-f(x)+\epsilon(h)$ where $\frac{\epsilon(h)}{h}\to0$ as $h\to0$.
This means that $f'(x)h$ is the best linear approximation of $\Delta y$ in a neighbourhood of x.
Now, you can also see $f'(x)h$ not as a number, but as a linear transformation in $h$. This is because we have linear approximations by a line in $\mathbb{R}^1$ whereas the natural generalisation of linear functions is linear transformations. 
Now, through the definition of the total derivative, the total derivative $A$ of $f$ at $x$ is the best approximation that satisfies 
$f(x+h)-f(x)=A(h)+\epsilon(h)$ where $\frac{|\epsilon(h)|}{|h|}\to0$ as $h\to0$.
That is $A(h)$ is the best linear approximation of $\Delta y$ in a neighborhood of $x$.
As for the partial derivatives, you can think of them like rates of change because all variables are fixed except one, that is you can think of the rate of change of the function as we move in one direction. 
I hope this helps.
A: Given a function, $f$, from $R^n$ to $R^m$, we can define the "derivative" of $f$, at a given point, to be the linear function, $y= Ax+ b$, from $R^n$ to $R^m$ that "best approximates" $f$.  In a given coordinate system for each of $R^n$ and $R^m$ that can be represented by an $m \times n$ matrix.  In particular, if $f$ is a "vector valued function of a single variable,  $f(t)= \langle a(t), b(t), c(d)\rangle$, then its derivative is $f'(t)= \langle a'(t), b'(t), c'(t)\rangle$.  If $f$ is a "real valued function of several variables, $f(x, y, z)$, its derivative is the gradient, $\nabla f= (\partial f/ \partial x, \partial f/\partial y, \partial f/ \partial z)$.
