
5-6 Calculus – Differentiation and Integration
= +
+
+8
3
3 2
21
2
cos x
x
x
d y
dx
x
x
x
d y
dx
x
x
2
2 2
3
3 3
8
9
3 2
42
8
54
3 2
42
= − −
+
+
= − +
+
+
sin
( )
cos
( )
.
Example 5.6 For the function defined by f(x) = x
2
+ 2x + 5, find the higher-order derivatives at x = 2.
Solution: Let y = f(x).
dx
dx
x x
x
dy
dx
x
= + +
= +
=
=
( )
.
2
2
2 5
2 2
6
d y
dx
d
dx
dy
dx
d
dx
x
d y
dx
x
2
2
2
2
2
2 2 2
=
= + =
=
( )
==
=
2
0
3
3
.
.
d y
dx
Therefore, the third derivative of the given function is zero.
Hence, other higher-order derivatives are also zero.
Example 5.7 Find the fifth-order derivative of the natural logarithm.
Solution: Given y = ln x.
dy
dx x
d y
dx x
d y
dx x
=
= −
=
1
1
2
2
2 2
3
3 3
.