13.3 Partial derivatives For an animation of this concept visit http://www.math.umn.edu/~rogness/multivar/dirderiv.shtml

When we have functions with more than one variable, we can find partial derivatives by holding all the variables but one constant. z 100 10 y 10 x Note: is also written as (eff sub ecks)

Notation for First Partial Derivatives

would give you the slope of the tangent in the plane y=0 or in any plane with constant y. z 100 10 y 10 x 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

Example 2 f(x,y) = e x y , find fx and fy 2 And evaluate each at the point (1,ln2) 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)