Download y , or f (x)

Calculus 12 – Ch. 2 Derivatives
Lesson 8: Higher Derivatives
 For any function y = f (x) , y! ,
(simply put: SLOPE)
dy
or f !(x) indicates the rate of change of y with respect to x
dx
 We can also take the derivative of the derivative
d2y
Notation:
y!!, f !!(x) or
dx 2
And taking the derivative again:
y!!!, f !!!(x) or
d 3y
dx 3
Etc.
 The second derivative is the rate of change of the slope of the tangent line with respect to x
 The second derivative of a position-time function represents acceleration (we will see more of
in Ch. 3)
Examples
1) Find y! and y!! for each of the following:
a) y = 4x + 3x ! 8
2
3
3
b) y = x ! 2 + x 2
x
1
y! = 8x + 3
1
1
1 !2
3
x + 6x !3 + x 2
2
2
1
6 3
=
+ 3+
x
2 x x 2
y" =
y!! = 8
3
y = x 2 ! 3x !2 + x 2
3
1
1 "
3 "
y!! = " x 2 "18x "4 + x 2
4
4
1
18
3
=" 3 " 4 +
x
4 x
4x 2
c) y = ( 2 ! x 2 )
10
d) y =
y = ( 3x + 4 )
y! = 10 ( 2 " x 2 ) ("2x )
9
= "20x ( 2 " x 2 )
y" = !
9
=!
(
) + "20x ( 9 )( 2 " x ) ( "2x )
= "20 ( 2 " x ) #( 2 " x ) " 18x %
$
&
= "20 ( 2 " x ) ( 2 " 19x )
y!! = "20 2 " x 2
1
3x + 4
9
2 8
2 8
2 1
2 8
2
y!! =
2
=
!
1
2
3
1
!
( 3x + 4 ) 2 (3)
2
3
3
2 ( 3x + 4 ) 2
5
9
"
( 3x + 4 ) 2 ( 3)
4
27
5
4 ( 3x + 4 ) 2
2) Use implicit differentiation to find y! and y!! for each of the following:
a) y 2 = x 2 + 2x
b) y 2 + 2y = 2x +1
2yy! + 2 y! = 2
2yy! = 2x + 2
y! =
y! ( y +1) = 1
x +1
y
y! =
1
y +1
y! = ( x + 1) y "1
y!! = y "1 + ( x + 1) ( "1) y "2 y!
= y + ( x + 1) ( "1) y
"1
=
1 ( x + 1)
"
y
y3
"2
( x + 1) y
2
y 2 " ( x + 1)
=
y3
y! = ( y +1)
"1
y!! = "1( y +1) y!
"2
= "1( y +1)
2
x 2 + 2x " (x 2 + 2x + 1)
y3
1
=" 3
y
=
"1
=
"1
( y +1)
3
"2
( y +1)
"1
c) x 3 + y 3 = 5
3x 2 + 3y 2 y! = 0
3y 2 y! = "3x 2
y! = "
x2
y2
y! = "x 2 y"2
y!! = "2xy"2 " x 2 ("2 ) y"3 y!
# x2 &
= "2xy"2 + 2x 2 y"3 %" 2 (
$ y '
="
=
=
=
2x 2x 4
" 5
y2
y
"2xy 3 " 2x 4
y5
"2x ( y 3 + x 3 )
y5
"10x
y5