Partial Derivatives

Definition of a Partial Derivative

Let f(x,y) be a function of two variables.  Then we define the partial derivatives  as

Definition of the Partial Derivative

if these limits exist.  

Algebraically, we can think of the partial derivative of a function with respect to x as the derivative of the function with y held constant.  Geometrically, the derivative with respect to x at a point P represents the slope of the curve that passes through P whose projection onto the xy plane is a horizontal line.  (If you travel due East, how steep are you climbing?)

Example

Let 

        f(x,y) = 2x + 3y 

then

We also use the notation f_x  and f_y for the partial derivatives with respect to x and y respectively.

Exercise:  

Find f_y for the function from the example above.

Finding Partial Derivatives the Easy Way

Since a partial derivative with respect to x is a derivative with the rest of the variables held constant, we can find the partial derivative by taking the regular derivative considering the rest of the variables as constants.

Example
   
Let 

        f(x,y)  =  3xy² - 2x²y

then 

        f_x  =  3y² - 4xy

and

        f_y  =  6xy - 2x²

Exercises

 Find both partial derivatives for

1. f(x,y) = xy sin x
   

2.                 x + y
   f(x,y) =                     
                   x - y

Higher Order Partials

Just as with function of one variable, we can define second derivatives for functions of two variables.  For functions of two variables, we have four types:

        f_xx,     f_xy     f_yx     and     f_yy

Example

Let 

        f(x,y)  =  y e^x  

then 

        f_x  =  ye^x

and

        f_y  =  e^x

Now taking the partials of each of these we get:

        f_xx = y e^x        f_xy = e^x        f_yx = e^x       and       f_yy = 0

Notice that    

        f_xy  =   f_yx  

Theorem   Let f(x,y) be a function with continuous second order derivatives, then                fxy  =   fyx

Functions of More Than Two Variables

Suppose that 

        f(x,y,z)  =  xy - 2yz 

is a function of three variables, then we can define the partial derivatives in much the same way as we defined the partial derivatives for three variables.  

We have

        f_x = y         f_y = x - 2z      and       f_z = -2y

Application:     The Heat Equation

Suppose that a building has a door open during a snowy day.  It can be shown that the equation

        H_t  =  c²H_xx     

models this situation where H is the heat of the room at the point x feet away from the door at time t.  Show that 

        H = e^-t cos(x/c) 

satisfies this differential equation.

Solution

We have

        H_t  =  -e^-t cos(x/c) 

        H_x  =  -1/c e^-t sin(x/c)

        H_xx  =  -1/c² e^-t cos(x/c)

So that 

        c²H_xx  =  -e^-t cos(x/c)

And the result follows.

Back to the Functions of Several Variables Page

Back to the Math 107 Home Page

Back to the Math Department Home Page

e-mail Questions and Suggestions