yx

z

10

10

100

An Introduction to Partial Derivatives

Greg Kelly, Hanford High School, Richland, Washington

yx

z

2 2, 100f x y x y

10

10

100

When we have functions with more than one variable, we can find partial derivatives by holding all the variables but one constant.

2df

xdx

2

22

d f

dx

2df

ydy

2

22

d f

dy

Note:df

dxis also written as xf (eff sub ecks)

yx

z

2 2, 100f x y x y

10

10

100

2df

xdx

2df

ydy

df

dxwould give you the slope of the tangent in the plane y=0 or in any plane with constant y.

In other words, how is changing one variable going to change the value of the function?

Mixed variables are also possible:

, cos xf x y x y ye

cos xdfy ye

dx

cos xd df dy ye

dy dx dy

2

sin

xd fy e

dy dx

sin xyxf y e

sin xdfx y e

dy

sin xd df dx y e

dx dy dx

2

sin

xd fy e

dx dy

sin xxyf y e

, cos xf x y x y ye

Both answers are the same!