The partial derivative extends the concept of the derivative in the one-dimensional case by studying real-valued functions defined on subsets of . Informally, the partial derivative of a scalar field may be thought of as the derivative of said function with respect to a single variable.
| Let , , be a function. Let denote the function value of at . We define the partial derivative of with respect to () to be the number
whenever the limit exists and is finite.
We thus associate with a function , called the partial derivative of with respect to the -th variable, where is the subset of where the limit above exists.
In it is common to write in place of , and we usually speak of the partial derivative of with respect to or , defined by
Example: dot product with fixed vector
Fix a vector , and define a function by
Then the partial derivative of with respect to is equal to :
We have shown that
This is an example of a property that can be shown to hold in general: when taking the partial derivative of a function with respect to some variable, one can differentiate as though all the other variables were constants in an ordinary derivative. That is, if we wish to compute the partial derivative of a function at a point with respect to , we may introduce another function given by , where the is in the -th place, and all the other components are held fixed. It is a trivial matter to verify that :
Example: two variables
Recall from the one-dimensional theory that if is any constant, then .
Now define a function by . The discussion above allows us to use the one-dimensional theory to compute and with ease: we simply note that in each of these derivatives, we may treat the second variable as a constant, and evaluate the derivative like in the one-dimensional case. Hence