Partial Differentiation

by AppliedMathematics.info

Printable Worksheet: Partial Differentiation

Partial differentiation has meaning in the context of the differentiation of functions of several variables. For example let f(x,y,z) be a function of x,y,z. The first order partial derivative of f with respect to x is deined as

f
x
 =
lim
h® 0 
 f(x+h,y,z)-f(x,y,z)
h
   ,
whenever the limit exists. The partial derivatives of f with respect to y and z are defined similarly. The partial derivatives
f
x
 , f
y
 , f
z
can be written fx, fy, fz.

By differentiating these functions again with respect to x, y, or z we obtain the higher order partial derivatives such as

2 f
x2
  ,        2 f
xy
  .

Chain Rule

Let F be a continuously differentiable function of f,g,h and suppose that each f,g,h is a continuously differentiable function of x,y,z. Then it may be shown that F is a continuously differentiable function of x,y,z and that

F
x
 = F
f
f
x
 + F
g
g
x
 + F
h
h
x
   ,

F
y
 = F
f
f
y
 + F
g
g
y
 + F
h
h
y
   ,

F
z
 = F
f
f
z
 + F
g
g
z
 + F
h
h
z
   ,
An equation formed from partial derivatives is called a partial differential equation .

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