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:

Similarly, let

be a function of three variables that is differentiable at

. The gradient of

Then by the theorem above, for any unit vector

For any unit vector

Remark: When

, all directional derivatives of

is zero. Geometric meaning of gradient Let

be a differentiable function of two variables and

be any unit vector. Then we have

where

is the angle between

and

. Then when will the directional derivative be the maximum/minimum?

has the greatest value when

(

). Therefore,

attains its maximum value when

i.e.

and

are in the same direction. Moreover, the maximum value of

has the smallest value when

(

. Therefore,

attains its minimum value when

i.e.

and

are in opposite directions. Moreover, the minimum value of

. Geometrically, the above results imply that when we consider point

and its corresponding point

on the graph of

, the slope of the curve in the direction of the gradient of

is the greatest i.e. the

is direction when the steepest ascent occurs at

. On the other hand,

is direction when the steepest descent occurs at

. In the applet below, you can check the checkbox "Gradient" to reveal the gradient of the function and its norm.

Example: Suppose

. Find the directions of the steepest ascent and descent on the graph of

when

. Moreover, find the maximum rate of the steepest ascent. Answer:

and

. Therefore, we have

, which is the direction of the steepest ascent. Moreover,

is the direction of the steepest descent. The maximum rate of ascent at

Exercise: Let . Find the maximum value of the directional derivative of at .

Gradient and level curves Suppose

is a differentiable function and

is any real number. Then the equation of the level curve of

when

. Suppose such level curve is parametrized by

. Then we have

Differentiate both sides and use chain rule, we have

Therefore,

is orthogonal to the tangent vector

of the level curve for all

. In the applet above, you can check the checkbox "Level curve" to reveal the level curve. You can see that the gradient vector is always normal to the level curve i.e. orthogonal to the tangent vector of the level curve.

Exercise: Let . Find the equation of line tangent to the level curve at .

Gradient and level surfaces Suppose

is a function of three variables such that its partial derivatives are all continuous (Note: This implies that

is differentiable). Let

be any real number. Then the equation of the level surface of

when

Consider any curve in the level surface, which is parametrized by

. Then we have

Differentiate both sides and use chain rule, we have

Therefore,

is orthogonal to the tangent vector at

for any

. Fix

and let

. Then

is orthogonal to the tangent vector of any curve on the level surface at

. In other words, if

is non-zero, then it is normal to the tangent plane of the level surface at

. Therefore, if

, the equation of the plane tangent to the level surface at

, or more explicitly,

In the applet below, the level surface

is shown. For any point

in the level surface, the gradient vector is normal to the tangent plane of the level surface at

Exercise: Find the equation of the tangent plane of a unit sphere centered at the origin at the point .

Directional Derivatives and the Gradient

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