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?)



        f(x,y) = 2x + 3y 



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


Find fy 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.


        f(x,y)  =  3xy2 - 2x2y


        fx  =  3y2 - 4xy


        fy  =  6xy - 2x2


 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:

        fxx,     fxy     fyx     and     fyy



        f(x,y)  =  y ex  


        fx  =  yex


        fy  =  ex

Now taking the partials of each of these we get:

        fxx = y ex        fxy = ex        fyx = ex       and       fyy = 0

Notice that    

        fxy  =   fyx  


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

        fx = y         fy = x - 2z      and       fz = -2y

Application:     The Heat Equation

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

        Ht  =  c2Hxx     

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.


We have

        Ht  =  -e-t cos(x/c) 

        Hx  =  -1/c e-t sin(x/c)

        Hxx  =  -1/c2 e-t cos(x/c)

So that 

        c2Hxx  =  -e-t cos(x/c)

And the result follows.


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