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
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
- 2 complex functions and the cauchy riemann equations columbia university
- graphing calculators and computers
- 4 1 exponential functions and their graphs michigan state university
- partial derivative of f with respect to x is denoted by
- dr matthew m conroy university of washington domain and range examples
- domain and range of composite functions calculator cÔng ty tnhh
- seismic trace interpolation in the f x ydomain imperial college london
- determine if a graph represents y as a function of x
- abstract algebra i auburn university
- guide to using the ti nspire for methods the simple and the
Related searches
- translate the graph of f x |3x| 5 down 6 units
- translate the graph of f x 3x 5 down 6 units
- derivative of arctan x a
- derivative of e x 2
- derivative of x squared
- derivative of e x 4
- derivative of fractions with variables
- derivative of ln x 2
- derivative of ln x 1
- derivative of cos x 2
- partial derivative calculator 3d
- partial derivative solver