Difference between revisions of "Partial derivative"
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===For a function of multiple variables=== | ===For a function of multiple variables=== | ||
− | {{ | + | The notation here gets a little messy, so read it carefully. We consider a function <math>f</math> of <math>n</math> variables, which we generically denote <math>(x_1,x_2,\dots,x_n)</matH> respectively. Consider a point <math>(a_1_a,2,\dots,a_n)</math> in the domain of the function. In other words, this is a point where <math>x_1 = a_1,x_2 =a_2, \dots, x_n = a_n</math>. |
+ | |||
+ | Suppose <math>i</math> is a natural number in the set <math>\{ 1,2,3,\dots,n \}</math>. | ||
+ | |||
+ | The '''partial derivative''' ''at'' this point <math>(a_1,a_2,\dots,a_n)</math> with respect to the variable <math>x_i</math> is defined as the [[derivative]]: | ||
+ | |||
+ | <math>\frac{d}{dx_i}f(a_1,a_2,\dots,a_{i-1},x_i,a_{i+1}, \dots,a_n)|_{x_i = a_i}</math> | ||
+ | |||
+ | In other words, it is the derivative (evaluated at <math>a_i</math>) of the function <math>x \mapsto f(x_1,x_2,\dots,x_{i-1},a_i,x_{i+1},\dots,x_n)</math> with respect to <math>x_i</math>, evaluated at the point <math>x_i = a_i</math>. |
Revision as of 00:26, 2 April 2012
Contents
Definition at a point
Generic definition
Suppose is a function of more than one variable, where is one of the input variables to . Fix a choice and fix the values of all the other variables. The partial derivative of with respect to , denoted , or , is defined as the derivative at of the function that sends to at for the same fixed choice of the other input variables.
For a function of two variables
Suppose is a real-valued function of two variables , i.e., the domain of is a subset of . We define the partial derivatives as follows:
- Partial derivative with respect to :
In words, it is the derivative at of the function .
This partial derivative is also denoted or .
- Partial derivative with respect to :
In words, it is the derivative at of the function .
This partial derivative is also denoted or .
For a function of multiple variables
The notation here gets a little messy, so read it carefully. We consider a function of variables, which we generically denote respectively. Consider a point Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): (a_1_a,2,\dots,a_n) in the domain of the function. In other words, this is a point where .
Suppose is a natural number in the set .
The partial derivative at this point with respect to the variable is defined as the derivative:
In other words, it is the derivative (evaluated at ) of the function with respect to , evaluated at the point .