Partial derivative

In practice there is nothing new to compute: to get \(f_x\) differentiate as in one variable treating \(y\) as a constant, and for \(f_y\) do the opposite. With \(f(x,y)=x^2y+3y^3\) one gets \(f_x=2xy\) and \(f_y=x^2+9y^2\). The notations \(\dfrac{\partial f}{\partial x}\) and \(\partial_x f\) are also used; the symbol \(\partial\) warns that other variables are held fixed.

Geometrically, slicing the surface \(z=f(x,y)\) with the vertical plane \(y=b\) produces a curve; \(f_x(a,b)\) is the slope of the tangent line to that curve at \(x=a\). Likewise \(f_y(a,b)\) is the slope in the direction of the \(y\)-axis. Like any derivative it measures a rate of change: if \(f_x(a,b)=3\), a small increase \(\Delta x\) in \(x\) changes \(f\) by approximately \(3\,\Delta x\).

Partial derivatives can be differentiated again, giving the second-order ones \(f_{xx}\), \(f_{yy}\) and the mixed ones \(f_{xy}\), \(f_{yx}\). If the latter are continuous they coincide (Clairaut's theorem), so for the usual functions the order does not matter. With \(f=x^2y^3\) one gets \(f_{xy}=f_{yx}=6xy^2\).

It is best not to over-interpret: the existence of \(f_x\) and \(f_y\) at a point only describes what happens along the two axes. For \(f(x,y)=|x|+y\), \(f_x(0,0)\) does not exist, and there are functions with both partials at a point that are not even continuous there. The notion that does control the behavior in every direction, differentiability, comes after the tangent plane.