Math 151 Section 3.8 Higher Order Derivatives f

( x )
 d
2 f dx
2
 d dx
( f ' ( x )) is the 2nd derivative of f ( x ).
The third derivative of
The nth derivative of f f
( x
( x ) is f
( 3 )
( x )

) is f
( n )
( x )
 d d
3 f n dx
3 dx f n

 d dx
( f " ( x )) d dx
( d n

1 dx n

1 f
)
Example : f ( x )
 x
5 f ' ( x )

5 x
4 f " ( x )

5

4 x
3 f
( 3 )
( x )

5

4

3 x
2 f
( n )
( x )

0 for n

6 f
( 4 )
( x )

5

4

3

2 x f
( 5 )
( x )

5

4

3

2

5 !
Examples : 1.
Find the derivative s of a ) f ( x )
 cos x b ) g ( x )
 sin x
2.
The height of an object is given by h ( t )

20

30 t

16 t
Find the velocity at time t , h ' ( t ), and the accelerati on, h " ( t ).
2 feet at t sec.
3 .
An object tra vels in a circle so the position at time t is given parametric ally by x ( t )

3 sin(

3 t ) y ( t )

3 cos(

3 t ).
 

3 rad/s is the angular ve locity.
Find the vector equations for the velocity and the accelerato n.
Compare the directions
 of r (t) and
 a (t) .

4 .
Find the nth derivative of each function.
a ) f(x)

1 x b ) f ( x )

1
1
 x c ) f ( x )
 xe x
5 .
Find the first and second derivative s.
a ) f ( x )
 x
2
1

1 b ) f ( x )
 sec x c ) f ( x )
 sin( x
2
)