Higher Order Derivatives

Objectives

Students will be able to• Calculate higher order derivatives• Apply higher order derivatives in

application problems

Symbol Representations

First Derivative

€

′ f (x) =d

dxf (x) =

dy

dx= ′ y

Second Derivative

€

′ ′ f (x) =d

dx

d

dxf (x)

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟=

d2f (x)

dx 2=

d2y

dx 2= ′ ′ y

Symbol Representations

Third Derivative

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′ ′ ′ f (x) =d

dx

d

dx

d

dxf (x)

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟=

d3f (x)

dx 3=

d3y

dx 3= ′ ′ ′ y

Fourth Derivative

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f (4 )(x) =d4f (x)

dx 4=

d4y

dx 4=y(4 )

Symbol Representations

nth Derivative

€

f (n)(x) =dnf (x)

dx n=

dny

dx n=y(n)

Example 1

Calculate the second derivative of the function

€

f (x) =−x 4 + 7x 3 −x 2

2

Example 2

For the function

find

€

′ ′ f (2)

€

′ ′ f (0)

€

f (x) =−x 4 + 7x 3 −x 2

2

Example 3

€

f (x) =−x

1−x 2

Calculate the second derivative of the function

Example 4

For the function

find

€

′ ′ f (8)

€

′ ′ f (0)

€

f (x) =−x

1−x 2

Example 5

€

f (x) = 2x 2 + 9

Calculate the second derivative of the function

Example 6

For the function

find

€

′ ′ f (9)

€

′ ′ f (0)

€

f (x) = 2x 2 + 9

Example 7

€

f (x) =−6x13

Calculate the second derivative of the function

Example 8

For the function

find

€

′ ′ f (2)

€

′ ′ f (0)

€

f (x) =−6x13

Example 9

Calculate the third and fourth derivative of the function

€

f (x) =2x 5 + 3x 4 −5x 3 + 9x −2

Example 10

Find the open interval(s) where the function is concave up or concave down. Find any points of inflection.

Example 11

Find the open interval(s) where the function is concave up or concave down. Find any points of inflection.

Example 12-1For an original function f(x) being a distance function with respect to time, the

first derivative of f(x) is the velocity (instantaneous rate of change of distance)

and the second derivative of f(x) is called acceleration (instantaneous rate of

change of velocity).

In terms of the demand

Example 12-2

A car rolls down a hill. Its distance (in feet) from its starting point is given by

where t is in seconds.

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f (x) =1.5t 2 + 4 t

a. How far will the car move in 10 seconds?

b. What is the velocity at 5 seconds? At 10 seconds?

c. How can you tell from v(t) that the car will not stop?

Example 12-3

A car rolls down a hill. Its distance (in feet) from its starting point is given by

where t is in seconds.

€

f (x) =1.5t 2 + 4 t

d. What is the acceleration at 5 seconds? At 10 seconds?

e. What is happening to the velocity and the acceleration as t increases?