Directional Derivatives and the Gradient

Directional derivatives Suppose

is a function of two variables and

is a point in the domain of

. Its partial derivatives

and

are the slope of the curve on the graph of

in the direction of x-axis and y-axis respectively. In fact, we can also consider the slope of the curve on the graph in any direction. Let

be a unit vector. We have the following definition: Definition: The directional derivative of

in the direction of

, denoted by

, is as follows:

(Note: By definition,

and

.) The applet below illustrates the geometric meaning of directional derivatives of a function of two variables. You can drag the point

and the red unit vector to see the change in the value of

For a function of three variables

, we can define its directional derivative in a similar way. Let

be a point in the domain of

and

be a unit vector in

. Then we have the following definition: Definition: The directional derivative of

in the direction of

The following useful theorem is a formula for computing directional derivatives: Theorem: Suppose

and

is a unit vector in

, then

For

and

is a unit vector in

, then

Proof: We only prove the result for functions of two variables here. The three variable version can be proved by similar argument. Let

and

. By chain rule, we have

Example: Let

and

. Find the directional derivative of

in the direction of

. Answer:

and

. Moreover, it can be easily checked that

is already a unit vector. (If the given vector is not a unit vector, we need to first normalize it into a unit vector pointing towards the same direction.)

Exercise: Let and . Find the directional derivative of at in the direction of .

Remark: For any unit vector

, it can be written as

, where

is the angle measured from the positive x-axis to the direction of

in anticlockwise direction. Therefore, we have the following formula for directional derivatives in terms of

Gradient Suppose

is a function of two variables that is differentiable at

. The gradient of

is the vector in

, denoted by

, is defined as follows:

Directional Derivatives and the Gradient

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