Difference between revisions of "Partial derivative"
From Calculus
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+ | Suppose <math>f</math> is a function of more than one variable, where <math>x</math> is one of the input variables to <math>f</math>. Fix a choice <math>x = x_0</math> and fix the values of all the other variables. The '''partial derivative''' of <math>f</math> with respect to <math>x</math>, denoted <math>\partial f/\partial x</math>, or <math>f_x</math>, is defined as the derivative at <math>x_0</math> of the function that sends <math>x</math> to <math>f</math> at <math>x</math> for the same fixed choice of the other input variables. | ||
===For a function of two variables=== | ===For a function of two variables=== |
Revision as of 22:56, 17 December 2011
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
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