TITLE: What is the best way to think about partial derivatives?
QUESTION [18 upvotes]: I'm looking for a good visual way to think about partial derivatives (and slopes and tangent lines of partial derivatives) since this concept is very new for me and  a little counter intuitive.
SO what is a good visual way to think about partial derivatives?
Thank you.

REPLY [4 votes]: The partial derivatices can also be seen as special cases of directional derivatives.
The definition of a directional derivative in the direction $v$ is given by:
$$
D_vf(x) = \lim_{h\rightarrow 0} \frac{f(x+hv)-f(x)}{h}
$$
If we now look at the partial derivatives with $f$ a function of two variables then
$$\frac{\partial f}{\partial x}=D_1f(x) = D_{(1,0)}f(x) =  \lim_{h\rightarrow 0} \frac{f(x+(h,0))-f(x)}{h}.$$ In case of the derivative in the $y$-direction we almost get the same:
$$
\frac{\partial f}{\partial y}=D_2f(x) = D_{(0,1)}f(x) =  \lim_{h\rightarrow 0} \frac{f(x+(0,h))-f(x)}{h}.
$$
So by studying this definitions we see that the partial derivative is the slope of one of the directions of the axis.