**Definition**: The derivative of *f* at *P*_{0}(*x*_{0}, *y*_{0})
in the direction of the unit vector u = u_{1}i + u_{2}j is
denoted by D_{u}*f*(*x*_{0}, *y*_{0}) and is the number

_{}

provided this limit exists. The definition for functions of three variables is entirely similar.

**Example**: Find the derivative of *f*(*x*, *y*) = *x*^{2} + *xy* at *P*_{0}(1,
2) in the direction _{}.

Evaluate *f* along the line through *P*_{0} in the direction u:

_{
}Differentiate with respect to the
parameter:

_{}

Evaluate:

_{}