Are partial derivatives a special case of the total derivative or just something else entirely? I can do basic multivariable calculations using partial and total derivatives. I also know for partial derivatives the existence of all partial derivatives at a point doesn't imply continuity. 
Are partial derivatives a special case of the total derivative or just something else entirely? Can someone compare / contrast them? 
 A: Here's something that may help:
If $x$ and $y$ are both independent functions of $t$ and $z=f(x,y)$ then
$$\frac{dz}{dt}=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}$$
In a sense, to get the total derivative, you add up all of the partial derivatives.
So to answer your question, the partial derivative isn't really a special case of the total derivative; it is, as bye_world pointed out in the comments, a $part$ of the total derivative.
A: The other responses answer your question well, but I would like to add a partial derivative is also a 'total' derivative of some function, which is the one-variable function obtained from your original function by setting all the other variables (different from the one with respect to which you are differentiating) constant. So, if, for concreteness, $f : \mathbb{R}^{2} \rightarrow \mathbb{R}$ and you wish to find the partial derivative w.r.t $x$ at $(x_{0},y_{0})$, then you can define $g(x) = f(x,y_{0})$ and then $g'(x_{0}) = \frac{\partial f}{\partial x} (x_{0},y_{0})$, and the l.h.s is a total derivative, whereas the r.h.s is partial derivative.
