Multivariate functions: Partial derivatives
Partial derivatives of the first order
When dealing with a bivariate function , you can keep fixed one of the two variables, say , and regard it as a constant. The result will be a function of a single variable . We study the derivative of this function of .
Partial derivative The (first order) partial derivatives of the bivariate function are the functions and defined by
The following four ways to write the partial derivative with respect to are commonly used: Similarly, for the partial derivative with respect to :
When we want to refer to the partial derivative of the function at some point , we use the following notation and Other common notation in this case is and .
The existence and many of the properties of the partial derivatives are not different from those of derivatives of functions of a single variable.
We formulate the definitions for the functions of two variables, but it will be clear how they can be defined for more than two variables.
The words "first order" are only important when we discuss higher partial derivatives; usually they are left out.
Why don't we write instead of ? Because in the case where and are functions of two variables, say and , the expressions and have different meanings. We will come back to this later.
Calculation rules for partial derivatives
The calculation rules for derivatives of functions of a single variable are applicable for partial derivatives. In particular, for the partial derivative with respect to , we have
Partial derivative with respect to : same rules, with replaced by .
The chain rule will be discussed later.
For functions of more than two variables, the partial derivatives can be defined in a similar manner.
We consider as a parameter and as the independent variable. So is constant.
The remainder of the formula depends on : it is . The derivative of it with respect to is equal to .
The end result is the product of the two intermediate results:
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