Functions of Two Variables
A function of two variables is usually written as , where
and
are the independent variables and
is the dependent variable. The domain of the function is the set of ordered pairs
and the set of corresponding values of
is called the range.
Partial Derivatives
If , then the partial derivatives with respect to
and
will be
(1)
and
(2)
respectively. In computing




Alternative Symbols
If , the partial derivatives can be represented using the following alternative symbols
(3)
and
(4)
High-Order and Mixed Derivatives and the Corresponding Alternative Symbols
If , the second-order partial derivative with respect to
will be
(5)
The third-order partial derivative with respect to

(6)
And, the mixed second-order partial derivative is
(7)
Chain Rule
Let’s say that , and
and
have continuous first-derivatives, then the chain rule is expressed as
(8)
and/or
(9)
8-Minute Lecture on Partial Derivatives
Reference
Dennis G. Zill. Advanced Engineering Mathematics, edition. Jones
Bartlett Learning. 2016.