Difference between revisions of "Partial derivative"
From Calculus
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* '''Partial derivative with respect to <math>y</math>''': | * '''Partial derivative with respect to <math>y</math>''': | ||
| − | <math>\frac{\partial f(x,y)}{\partial y}|_{(x,y) = (x_0,y_0)} = \frac{d}{ | + | <math>\frac{\partial f(x,y)}{\partial y}|_{(x,y) = (x_0,y_0)} = \frac{d}{dy}f(x_0,y)|_{y = y_0}</math> |
In words, it is the [[derivative]] at <math>y = y_0</math> of the function <math>y \mapsto f(x_0,y)</math>. | In words, it is the [[derivative]] at <math>y = y_0</math> of the function <math>y \mapsto f(x_0,y)</math>. | ||
Revision as of 00:12, 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
Fill this in later
