13.3 Partial derivativesFor an animation of this concept visit

http://www.math.umn.edu/~rogness/multivar/dirderiv.shtml

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)

Notation for First Partial Derivatives

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?

Definition of Partial Derivatives of a Function of Two Variables

f(x,y) = e x y , find fx and fy And evaluate each at the point (1,ln2)

2

Example 2

Diagram for example 2

Example 2 solution

Example 3

Find the slope in the x-direction and in the

y-direction of the surface given by

When x=1 and y=2

Solution to example 3

Example 4

Find the slope of the given surface in the

x-direction and the y-direction at the point (1,2,1)