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

fx = 3y2 - 4xy

Similarly

fy = 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:

fxx,     fxy,     fyx    and     fyy

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) = yex

then

fx = yex

and

fy = ex

Now taking the partials of each of these we get:

fxx = yex,     fxy = ex,     fyx = ex,    and    fyy = 0

Notice that

fxy =  fyx

 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

fx = y ,   fy = x - 2z   and   fz = -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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