partial derivative:

partial derivative:
Functions of Several Variables:
of several independent real variables are defined similarly to functions in the single-variable case.
Suppose D is a set of n-tuples of real numbers (x_1 ?,x?_2,…,x_n). A real-valued function f on D is a rule that assigns a unique (single) real number
w = f(x_1 ?,x?_2,…,x_n)
Real-valued functions
to each element in D.
The set D is the function s domain. The set of w-valuestaken on by f is the function s range.
The symbol w is the dependent variable of f, and fis said to be a function of the n independent variablesx_1tox_n.
Also thex_j’s are called the function s input variables, and w is called the function s output variable.
Example: The value of f(x,y)=x^2+?2y?^3 at the point (3,-1) is:
f(3,-1)=?(3)?^2+?2(-1)?^3=9-2=7.
Example: The value of f(x,y,z)=?(x^2+y^2+z^2 ) at the point (3,0,4) is:
f(3,0,4)=?((3)^2+(0)^2+(4)^2 )=?(9+0+16)=?25=5.
Partial Derivatives
The ordinary derivative of a function of several variables with respect to one of the independent variables, keeping all other independent variables constant, is called the partial derivative of the function with respect to the variable.
? Partial derivatives of f(x,y)with respect to x is denoted by ?f/?x (orf_x,f_x (x,y) ).
The partial derivative of f(x,y) with respect to x at the point (x_0,y_0) is
? ?f/?x?|_((x_0,y_0))=? d/dx f(x,y_0 ) ?|_(x_0=0)=lim?(h?(?0))??(f(x_0+h,y_0 )-f(x_0,y_0 ))/h?
provided the limit exists.
? Partial derivatives of f(x,y)with respect to y is denoted by ?f/?y (orf_y,f_y (x,y) ).
The partial derivative of f(x,y) with respect to y at the point (x_0,y_0) is