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Understanding Directional Derivatives and the Maximum Rate of Change, Slides of Advanced Calculus

This lecture explains how to find the rate of change of a multivariable function in the direction of an arbitrary vector using the concept of directional derivatives. The lecture also discusses the maximum rate of change and the direction in which it occurs.

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16. LECTURE 16
I understand how to find the rate of change in any direction.
I understand in what direction the maximum rate of change happens.
Objectives
So far, we’ve learned the definition of the gradient vector and we know that it tells us the direc-
tion of steepest ascent. Its magnitude indicates the rate of change of the dependent variable in the
direction of the gradient.
A natural question to ask is, “What is the rate of change of the dependent variable in the direction
of an arbitrary vector ~v?” In other words, how fast does the surface ascend in the direction of ~v?
The rate of change in the direction of ~vis called the directional derivative. That is the scalar
projection of the gradient onto ~v.
Definition 16.1: The directional derivative, denoted Dvf(x, y), is a derivative of a multivariable
function in the direction of a vector ~v. It is the scalar projection of the gradient onto ~v.
Dvf(x, y)=compvrf(x, y)= rf(x, y)·~v
|~v|
It’s best to understand concepts with a picture. So let’s draw one. Consider the function
f(x, y)=x2y2.
The gradient of fis
rf(x, y)=2x
2y.
At the point (1,0), the direction of steepest ascent is
rf(1,0) = 2
0.
In that direction, fhas a slope of |rf(1,0)|=p(2)2=2.
91
pf3
pf4
pf5

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16. L ECTURE 16

⇤ I understand how to find the rate of change in any direction.

⇤ I understand in what direction the maximum rate of change happens.

Objectives

So far, we’ve learned the definition of the gradient vector and we know that it tells us the direc-

tion of steepest ascent. Its magnitude indicates the rate of change of the dependent variable in the

direction of the gradient.

A natural question to ask is, “What is the rate of change of the dependent variable in the direction

of an arbitrary vector ~v?” In other words, how fast does the surface ascend in the direction of ~v?

The rate of change in the direction of ~v is called the directional derivative. That is the scalar

projection of the gradient onto ~v.

Definition 16.1: The directional derivative, denoted Dv f (x, y), is a derivative of a multivariable

function in the direction of a vector ~v. It is the scalar projection of the gradient onto ~v.

Dv f (x, y) = compv rf (x, y) =

rf (x, y) · ~v

|~v|

It’s best to understand concepts with a picture. So let’s draw one. Consider the function

f (x, y) = x

2 y

2 .

The gradient of f is

rf (x, y) =

2 x

2 y

At the point (1, 0), the direction of steepest ascent is

rf (1, 0) =

In that direction, f has a slope of |rf (1, 0)| =

p (2) 2 = 2.

What is the slope at (1, 0) in the direction of

~v =

D (^) v f (x, y) = comp v

rf (1, 0) =

p 0

2

  • 1

2

Let’s look at some examples.

First, we find the partial derivatives to define the gradient.

f (^) x (x, y, z) =

yz

p xyz

f (^) y (x, y, z) =

xz

p xyz

f (^) z (x, y, z) =

xy

p xyz

The gradient is

rf (3, 2 , 6) =

0 B B B B @

12 2(6)

18 2(6)

6 2(6)

1 C C C C A

0 B B B B @

3 2

1 2

1 C C C C A

A

The magnitude of ~v =

A (^) is

|~v| =

p 1 + 4 + 4 = 3

Therefore, the directional derivative is

Dv f (3, 2 , 6) =

rf (3, 2 , 6) · ~v

|~v|

A ·

A =

The next natural question is:

In what direction is the derivative maximum?

As we just saw, the directional derivative is calculated by taking the scalar projection of rf onto

a vector ~v. Define ✓ be the angle between ~v and rf. Then,

rf · ~v

|~v|

|rf ||~u| cos(✓)

|~v|

= |rf | cos(✓)

This is maximized if ✓ = 0. From this, we know the following:

  • The maximum rate of change (the largest directional derivative) is |rf |.
  • This occurs when ~v is parallel to rf , i.e. when they point in the same direction.

That makes sense since rf is the vector pointing toward steepest ascent, so it should be the

direction with the largest derivative.

Observe, also that...

  • No change occurs when ✓ = 90

or when ✓ = 90

. In other words, directions perpendic-

ular to the gradient are constant height.

  • The rate of steepest descent happens when ✓ = 180

. It’s rate of change is |rf |,

Let’s look at two examples.

Example 16.4: Find the maximum rate of change of f at the given point and the direction in which

it occurs.

f (s, t) = te

st , at (0, 2)

The maximum rate of change is |rf (0, 2)|. Let’s first find the gradient.

rf =

te

st

ste

st

  • e

st

Then

|rf (0, 2)| =

p

(2)

2

  • 1

2

p

5

The direction is

rf (0, 2) =

Remark 16.5: For this problem, it may not have been clear which component was the first and

which was the second since s and t are atypical variables. For clues about the order, look at how

the ordered pairs are defined in the function. It was written as “f (s, t),” which tells us our gradient

vector should be

f (^) s

f (^) t

Example 16.6: Find the maximum rate of change of f at the given point and the direction in which

it occurs.

f (x, y, z) =

p

x 2 + y 2 + z 2 , (3, 6 , 2)

As above, the maximum rate of change is |rf (3, 6 , 2)|.

rf (x, y, z) =

0 B B B B B B @

p x x 2 +y 2 +z 2

y p x 2 +y 2 +z 2

z p x 2 +y 2 +z 2

1 C C C C C C A

Then

rf (3, 6 , 2) =

0 B B B B @

3 7

6 7

2 7

1 C C C C A

The |rf (3, 6 , 2)| = 1/ 7

p 9 + 36 + 4 = 1