Definition of a Partial Derivative
Let f(x,y) be a function of two variables. Then we define the partial
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
the slope of the curve that passes through P whose projection onto the
plane is a horizontal line. (If you travel due East, how steep are
We also use the notation fx
for the partial derivatives with respect to x and y respectively.
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.
Find both partial derivatives for
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:
Now taking the partials of each of these we get:
Functions of More Than Two Variables
Application: The Heat Equation
models this situation where H is the heat of the room at the point
away from the door at time t. Show that