Directional Derivatives and the Gradient

(1) Definition (Directional Derivative) Let be a function of two variables, and let be a unit vector. The directional derivative of at in the direction of is given by

provided the limit exists.

(2) Proposition (Directional Derivative) Let be a function that is differentiable at Then has a directional derivative in the direction of the unit vector which is given by

(3) Example (Directional Derivative) Find the directional derivative if

and is the unit vector given by the angle What is

    Solution. We have, and

Therefore,

(4) Example (Directional Derivative) Let be defined by Find the directional derivative of at in the direction toward the origin.

Solution. A vector in the direction from to is so a unit vector in this direction is therefore We find that

and

Therefore, the directional derivative is given by

(5) Example (Directional Derivative) Let be defined by Find the directional derivative of at in the direction toward the origin.

Solution. A vector in the direction from to is so a unit vector in this direction is therefore We find that

and

Therefore, the directional derivative is given by

(6) Definition (Gradient) Let be a differentiable function at and let have partial derivatives and Then the gradient of is denoted by

The value of the gradient at the point is denoted by

(7) Proposition (Gradient) Let and be differentiable functions. Then

        (i) for any constant

        (ii) for any constants and

        (iii)

        (iv)   provided

        (v)


(8) Proof. The Constant Rule is proved as follows:



The Linearity Rule is proved as follows:

The Product Rule is proved as follows:

The Quotient Rule is proved as follows:

The Power Rule is proved as follows: