A. For , the partial derivative of f with respect to x is

Since y is unchanging in this definition, is computed by treating y as a constant.

With that in mind find if . Check your answer on your TI-89.

Other notation for include

Partial derivative may be interpreted as slope of tangent line to the surface of determined by the trace y = b.

 

It is the slope at (a,b) in "the x-direction."

is the instantaneous rate of change of f with respect to x.

You can also find all of the above for .

Similarly, you can define and find partial derivatives for function of more than two variables.

Implicit partial differentiation is preformed in the same manner as was done for functions of one variable. Remember to treat the other variables as constants.

PROBLEMS

1. Find and for

2. Find and for

3.  Find for

4.  Find the sloop of the tangent line in the y direction to

at the point ( 1, 2, 13 ).

5. The level curves for are given below:

Is positive or negative? What about ?

6. Find where is given by .

B. Second partial derivatives

For a function there are four second partial derivatives:

For example, is found by finding the partial derivative of f with respect to x and taking the derivative of that result with respect to y.

Clairaut’s Theorem: (For some functions two of these four are the same.) If f is defined on a disk containing (a, b) and both f_xy and f_yx are continuous on that disk, then f_xy = f_yx at (a,b).

PROBLEMS

7. Find for

8. For , is for all (x,y)?

9.  True or False:

If all third partial derivatives are continuous, then

a)                               b)

10.  Let

a)  Find f_xyy

b) Find f_yxy