TURUNAN ORDE TINGGIYUSRON SUGIARTO

Slide 1.8- 2Copyright © 2008 Pearson Education, Inc.

Publishing as Pearson Addison-Wesley

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Higher Order Derivatives

OBJECTIVES

Find derivatives of higher order.

Given a formula for distance, find velocity and acceleration.

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Consider the function given by

Its derivative f is given by

The derivative function f can also be differentiated. We can think of the derivative f as the rate of change of the slope of the tangent lines of f . It can also be regarded as the rate at which is changing.

y f (x) x5 3x4 x.

y f (x) 5x4 12x3 1.

f x

HIGHER-ORDER DERIVATIVES

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Higher-Order Derivatives (continued):

We use the notation f for the derivative .

That is,

We call f the second derivative of f. For

the second derivative is given by

f (x) d

dxf (x)

y f (x) x5 3x4 x,

y f (x) 20x3 36x2 .

f

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Higher-Order Derivatives (continued):

Continuing in this manner, we have

When notation like gets lengthy, we abbreviate

it using a symbol in parentheses. Thus is the

nth derivative.

f (x) 60x2 72x, the third derivative of f

f (x) 120x 72, the fourth derivative of f

f (x) 120, the fifth derivative of f .

f x fn x

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Higher-Order Derivatives (continued):

For we havey f (x) x5 3x4 x,

f (3)(x) 60x2 72x,

f (4 )(x) 120x 72,

f (5)(x) 120,

f (6)(x) 0, and

f (n)(x) 0, for any integer n 6.

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Higher-Order Derivatives (continued):

Leibniz’s notation for the second derivative of a

function given by y = f(x) is

read “the second derivative of y with respect to x.”

The 2’s in this notation are NOT exponents.

d2y

dx2, or

d

dx

dy

dx

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Higher-Order Derivatives (concluded):

If then y x5 3x4 x,

dy

dx 5x4 12x3 1,

d 4y

dx4 120x 72,

d 2y

dx2 20x3 36x2 ,

d 5y

dx5 120.

d 3y

dx3 60x2 72x,

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Excercise 1: For find d 2y

dx2.y

1

x,

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Excercise 2: For find and .y (x2 10x)20 , y y

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DEFINITION:

The velocity of an object that is s(t) units from a

starting point at time t is given by

Velocity v(t) s (t) limh0

s(t h) s(t)

h

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DEFINITION:

Acceleration a(t) v (t) s (t).

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Excercise 4: For s(t) = 10t2 find v(t) and a(t), where s

is the distance from the starting point, in miles, and t is

in hours. Then, find the distance, velocity, and

acceleration when t = 4 hr.

TERIMA KASIHYUSRON SUGIARTO