16.3 Partial differentiation
The lines in the tangent plane parallel to thexz-plane, give the slope of the surface in the
x-direction at each point. When considering rate of change in thex-direction alone, the
y-coordinate inz=f(x,y)can be regarded as constant. Then for this fixed valued ofy
z=f(x,y)is a function ofx, a curve parallel to thexzplane. The slope of this curve can be obtained
in the usual way by differentiating:
dz
dx=df
dxHowever, this notation is not appropriate becausef is now a function of two variables,
so we use:
∂z
∂x=∂f
∂x(x, y)read ‘curly dee dee ex’.
This is called thepartial derivative ofzwith respect tox, and represents the rate of
change ofzwith respect toxassumingyis constant. We will also denote it byfx.Itgives
the rate of change inzin thex-direction, or the slope of the surface in thex-direction
(see Figure 16.5). We can define the partial derivative rigorously in terms of a limit as
follows:
∂f
∂x=lim
h→ 0[
f(x+h, y)−f(x,y)
h]Similarly, we can define the partial derivative with respect toy:
∂f
∂y=lim
k→ 0[
f(x,y+k)−f(x,y)
k]which is the derivative off with respect toyassumingxconstant. We will sometimes
usefy. It gives the rate of change off in they-direction, or the slope of the surface in
they-direction.
The rules of partial differentiation are the same as those in ordinary differentiation, so
long as we remember which of the variables to hold constant. Specifically (234
➤
):
∂
∂x(f±g)=∂f
∂x±∂g
∂x
∂
∂x(fg)=f∂g
∂x+∂f
∂xg∂
∂x(
f
g)
=g∂f
∂x−f∂g
∂x
(g)^2
∂f
∂x(g(x), y)=∂g
∂x∂f
∂g(g, y)