y x z 10 100 an introduction to partial derivatives greg kelly, hanford high school, richland,...
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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!