Partial derivative of f with respect to x is denoted by

PARTIAL DERIVATIVES

Notation and Terminology: given a function f (x, y) ;

? partial derivative of f with respect to x is denoted by

f

(x, x

y)

fx(x,

y)

Dxf

(x,

y)

f1;

? partial derivative of f with respect to y is denoted by

f y (x, y) fy(x, y) Dyf (x, y) f2.

Definitions: given a function f (x, y);

? definition for fx(x, y):

f (x + h, y) - f (x, y)

fx(x, y) = lim

h0

h

;

? definition for fy(x, y):

f (x, y + h) - f (x, y)

fy(x, y) = lim

h0

h

.

Determination of fx and fy:

? to find fx(x, y): keeping y constant, take x derivative; ? to find fy(x, y): keeping x constant, take y derivative.

Graphical Interpretation of fx and fy:

? fx(a, b) is slope of tangent line in x direction for the surface z = f (x, y) at f (a, b);

? fy(a, b) is slope of tangent line in y direction for the surface z = f (x, y) at f (a, b).

PARTIAL DERIVATIVES CONTINUED

Applications to Implicit Differentiation

Functions of n > 2 Variables: given f (x) = f (x1, . . . , xn)

? notation and terminology: the

partial derivative of f with respect to xi is denoted by

f

(x) xi

fxi(x)

Dxif

(x)

fi(x);

definition:

fxi(x)

=

lim

h0

f

(x1,

.

.

.

,

xi-1,

xi

+

h, h

xi+1,

.

.

.

,

xn)

-

f

(x) .

Higher Derivatives: given f (x, y) defined on a domain D;

? notation: second partials are denoted by

2f

f

x2 (x,

y)

() x x

(fx)x

f11;

2f

f

(x, yx

y)

() y x

fxy

(fx)y

f12;

2f

f

(x, xy

y)

( x y

)

fyx

(fy)x

f21;

2f

f

y2

(x,

y)

() y y

(fy)y

f22.

? similar notation for functions with > 2 variables.

? Clairaut's Theorem: if (a, b) D, and fxy and fyx are both continuous on D, then

fxy(a, b) = fyx(a, b).

? Applications to Partial Differential Equations

2

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