There are some special unit vectors which come to mind immediately. These are the vectors e_{i} where

and the 1 is in the i^{th} position. The partial derivatives are simply directional derivatives taken in these special directions.
Definition 20.3.3 Let U be an open subset of ℝ^{n} and let f : U → ℝ. Then letting x =

This is called the partial derivative of f. Thus,
Example 20.3.4 Find
From the definition above,

Also observe that

Higher order partial derivatives are defined by analogy to the above. Thus in the above example,

These partial derivatives, f_{xy} are called mixed partial derivatives.
There is an interesting relationship between the directional derivatives and the partial derivatives, provided the partial derivatives exist and are continuous.
Definition 20.3.5 Suppose f : U ⊆ ℝ^{n} → ℝ where U is an open set and the partial derivatives of f all exist and are continuous on U. Under these conditions, define the gradient of f denoted ∇f

This proposition will be proved in a more general setting later. For now, you can use it to compute directional derivatives.
Example 20.3.7 Find the directional derivative of the function

at
First find the gradient.

Therefore,

The directional derivative is therefore,

Another important observation is that the gradient gives the direction in which the function changes most rapidly. The following proposition will be proved later.
Proposition 20.3.8 In the situation of Definition 20.3.5, suppose ∇f
The concept of a directional derivative for a vector valued function is also easy to define although the geometric significance expressed in pictures is not.
Definition 20.3.9 Let f : U → ℝ^{p} where U is an open set in ℝ^{n} and let v be a unit vector. For x ∈ U, define the directional derivative of f in the direction v, at the point x as

Example 20.3.10 Let f
First, a unit vector in this direction is



You see from this example and the above definition that all you have to do is to form the vector which is obtained by replacing each component of the vector with its directional derivative. In particular, you can take partial derivatives of vector valued functions and use the same notation.
Example 20.3.11 Find the partial derivative with respect to x of the function f
From the above definition, f_{x}