Graphical representation of derivatives in higher dimensions than 2D... I know how the derivative of y with respect to x ($\frac{dy}{dx}$)  represents the slope of the the line tangent to the curve at the point with x-coordinate as 'x', in the x-y plane ; this is when we speak of 2D space.
What I wish to know is what these derivatives represent in higher dimensions.
Eg. in 3D, what does the value of $f'(x,y)$ represent? 
Take in mind I am still in high school so explain in a reasonably understandable language.
Thanks..
 A: I would say prime notation is a bit out of its depth here. If you are taking the derivative of a multivariate function $f(x,y)$, you would need to specify the variable with respect to which you are differentiating. 
For example, letting $f(x,y)=x+y$:
$$\frac{\partial}{\partial x}f(x,y)=y+1$$
$$\frac{\partial}{\partial y}f(x,y)=x+1$$
using Leibniz notation. If you haven't seen this notation, the choice of $x$ or $y$ in the denominator of the $\frac{\partial}{\partial x}$ tells the reader that you are not interested in the other variable.  I have never seen the notation $f'(x,y)$ used, and I don't think it would be considered acceptable by most authors. Hope this helps!
P.S. I'm in my final year of high school, so hopefully this isn't hard to understand :)
A: The gradient of a function is given by $$\nabla f(x,y)=\begin{pmatrix}\frac{\partial f}{\partial x}\\\frac{\partial f}{\partial y}\end{pmatrix}$$
The geometrical interpretation is that this is a vector pointed in the direction where you have the steepest increase. The magnitude is the slope in that direction. See for example the answers to this question
