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?)

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 we can think of the function as 

        f(x,y)  =  (3y²) x  -  (2y) x²

        =  (3y²) (1)  -  (2y) (2x)

so that

        f_x = 3y² - 4xy

Similarly

        f_y = 6xy - 2x²

Exercises

 Find both partial derivatives for

1. f(x,y) = xysinx
   

2. f(x,y) = (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. 

since we can 

1. First take a partial derivative with respect to x and then again with respect to x.
   

2. First take a partial derivative with respect to x and then again with respect to y.
   

3. First take a partial derivative with respect to y and then again with respect to x.
   

4. First take a partial derivative with respect to y and then again with respect to y.

Example

Let 

        f(x,y) = ye^x  

then 

        f_x = ye^x

and

        f_y = e^x

Now taking the partials of each of these we get:

        f_xx = ye^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:  Marginal Cost

A rental shop rents out bicycles and mopeds. The cost function is given by 

where x represents the number of bicycles rented and y represents the number of mopeds rented.  Find the marginal cost with respect to the number of mopeds sold when x = 36 and y = 25.

Solution

This is just the partial derivative with respect to y.  Since

  We get

 Now plugging in (36, 25) gives

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