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# First partial derivatives.

Let f be a function of two variables x,y, defined on a neighborhood of the point P0=(x0,y0).

Definition 5.1

• The first partial derivative of f at P0 with respect to the variable x is

provided the limit exists and is finite.
• The first partial derivative of f at P0 with respect to the variable y is

provided the limit exists and is finite.

For a function of more than two variables, we define the other partail derivatives by the same way.

In each case, we consider one variable as a constant and differentiate with respect to the other variable, according to the well-known rules of differentiation, as learnt in Calculus I.

Example 5.2   Let . Then at any point , we have:

Example 5.3   Let . Then at any point , we have:

Example 5.4   Find when z is defined as an implicit function of x and y by the relation .

We differentiate both sides of the relation with respect to x:

Notations: and .

Next: Differentiability, linearization. Up: Partial derivatives; the differential Previous: Partial derivatives; the differential
Noah Dana-Picard
2001-05-30