1. Partial derivative b a

2. Let be defined on and let Define If has a derivative at we call its value the partial derivative of by at

3. We use several different ways to denote the partial derivative of by at : Note that the symbol reads "d" and is not identical to the Greeek delta !!!

4. The function in n variables that assigns to each at which the partial derivative of exists the value of such a partial derivative is then called the partial derivative of by x i Sometimes this function is also denoted

5. When calculating such a partrial derivative, we use the following practical approach: When calculating the partial derivative of by x i, we think of every variable other than x i as of a constant parameter and treat it as such. Then we actually calculate the "ordinary" derivative of a function in one variable.

6. Calculate all the partial derivatives of the following functions

7. Partial derivatives of higher orders If a partial derivative is viewed as a function it may again be differentiated by the same or by a different variable to become a partial derivative of a higher order. Theoretically, there may be a partial derivative of an arbirary order if it exists. Notation:

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9. Schwartz' theorem Let be an internal point of the domain of and let, in a neighbourhood of X, exist and be continuus at. Then exists and Similar assertions also hold for functions in more than two variables and for higher order derivatives.