Section 15.3 Partial Derivatives

PARTIAL DERIVATIVES If f is a function of two variables, its partial derivatives are the functions f x and f y defined by

NOTATIONS FOR PARTIAL DERIVATIVES If z = f (x, y), we write

RULE FOR FINDING PARTIAL DERIVATIVES OF z = f (x, y) 1.To find f x, regard y as a constant and differentiate f (x, y) with respect to x. 2.To find f y, regard x as a constant and differentiate f (x, y) with respect to y.

FUNCTIONS OF MORE THAN TWO VARIABLES There are analogous definitions for the partial derivatives of functions of three or more variables.

GEOMETRIC INTERPRETATION OF PARTIAL DERIVATIVES Consider the surface S whose equation is z = f (x, y). The plane y = b intersects this surface in a plane curve C 1. The value of f x (a, b) is the slope of the tangent line T 1 to the curve at the point P(a, b, f(a, b)). Similarly, the plane x = a intersects the surface in a plane curve C 2 and f y (a, b) is the slope of the tangent line T 2 to the curve at the point P(a, b, f(a, b)).

SECOND PARTIAL DERIVATIVES If f is a function of two variables, then its partial derivatives f x and f y are also functions of two variables, so we can consider their partial derivatives ( f x ) x, ( f x ) y, ( f y ) x, and ( f y ) y, which are called the second partial derivatives of f.

NOTATION FOR THE SECOND PARTIAL DERIVATIVES If z = f (x, y), we use the following notation:

CLAIRAUT’S THEOREM Suppose that f is defined on a disk D that contains the point (a, b). If the functions f xy and f yx are both continuous on D, then f xy (a, b) = f yx (a, b)

HIGHER ORDER DERIVATIVES If f is a two variable function, partial derivatives of order 3 and higher can be defined. Some examples would be f xxx, f xyx, f xyyx, etc. Using Clairnaut’s Theorem, we can show that f xyy = f yxy = f yyx if these functions are continuous and f xxy = f xyx = f yxx if these functions are continuous.