This chapter discusses differential calculus, including definitions of derivatives, differentiation, and the derivative. It provides examples of calculating derivatives of various functions using basic differentiation rules and the chain rule. Higher-order derivatives and applications to finding maxima and minima are also covered. Key concepts are defined, such as the derivative representing the slope of a function and differentiation being the process of obtaining a derivative. Formulas for common derivatives are presented.

1. 1
Chapter 6
Differential Calculus
The two basic forms of calculus are
differential calculus and integral calculus.
This chapter will be devoted to the former
and Chapter 7 will be devoted to the
latter. Finally, Chapter 8 will be devoted
to a study of how MATLAB can be used for
calculus operations.

2. 2
Differentiation and the Derivative
The study of calculus usually begins with
the basic definition of a derivative. A
derivative is obtained through the process
of differentiation, and the study of all forms
of differentiation is collectively referred to
as differential calculus.If we begin with a
function and determine its derivative, we
arrive at a new function called the first
derivative. If we differentiate the first
derivative, we arrive at a new function
called the second derivative, and so on.

3. 3
The derivative of a function is the
slope at a given point.

4. 4
Various Symbols for the Derivative
( )
or '( ) or
dy df x
f x
dx dx
0
Definition: lim
x
dy y
dx x∆ →
∆
=
∆

5. 5
Figure 6-2(a). Piecewise Linear
Function (Continuous).

6. 6
Figure 6-2(b). Piecewise Linear
Function (Finite Discontinuities).

7. 7
Piecewise Linear Segment

8. 8
Slope of a Piecewise Linear Segment
2 1
2 1
slope
y ydy
dx x x
−
= =
−

9. 9
Example 6-1. Plot the first derivative
of the function shown below.

10. 10

11. 11
Development of a Simple Derivative
2
y x=
2
( )y y x x+ ∆ = + ∆
2 2
2 ( )y y x x x x+ ∆ = + ∆ + ∆

12. 12
Development of a Simple Derivative
Continuation
2
2 ( )y x x x∆ = ∆ + ∆
2
y
x x
x
∆
= + ∆
∆
0
lim 2
x
dy y
x
dx x∆ →
∆
= =
∆

13. 13
Chain Rule
( )y f u= ( )u u x=
( )
'( )
dy df u du du
f u
dx du dx dx
= =
( )
'( )
df u
f u
du
=where

14. 14
Example 6-2. Approximate the derivative
of y=x2
at x=1 by forming small changes.
2
(1) (1) 1y = =
2
(1.01) (1.01) 1.0201y = =
1.0201 1 0.0201y∆ = − =
0.0201
2.01
0.01
dy y
dx x
∆
≈ = =
∆

15. 15
Example 6-3. The derivative of sin u
with respect to u is given below.
( )sin cos
d
u u
du
=
Use the chain rule to find the
derivative with respect to x of
2
4siny x=

16. 16
Example 6-3. Continuation.
2
u x=
2
du
x
dx
=
2
'( )
4(cos )(2 ) 8 cos
dy du dy du
f u
dx dx du dx
u x x x
= =
= =

17. 17
Table 6-1. Derivatives
( )f x '( )f x Derivative Number
( )af x '( )af x D-1
( ) ( )u x v x+ '( ) '( )u x v x+ D-2
( )f u ( )
'( )
du df u du
f u
dx du dx
=
D-3
a 0 D-4
( 0)n
x n ≠ 1n
nx − D-5
( 0)n
u n ≠ 1n du
nu
dx
−
D-6
uv dv du
u v
dx dx
+ D-7
u
v
2
du dv
v u
dx dx
v
− D-8
u
e u du
e
dx
D-9

18. 18
Table 6-1. Derivatives (Continued)
u
a
( )ln u du
a a
dx
D-10
lnu 1 du
u dx
D-11
loga u
( )
1
loga
du
e
u dx
D-12
sinu
cos
du
u
dx
 
 
 
D-13
cosu
sin
du
u
dx
− D-14
tanu 2
sec
du
u
dx
D-15
1
sin u−
1
2
1
sin
2 21
du
u
dxu
π π− 
− ≤ ≤ 
 −
D-16
1
cos u−
2
1
1
du
dxu
−
−
( )1
0 cos u π−
≤ ≤ D-17
1
tan u−
1
2
1
tan
1 2 2
du
u
u dx
π π− 
− < < 
+  
D-18

19. 19
Example 6-4. Determine dy/dx for
the function shown below.
2
siny x x=
( ) ( )2
2 sin
sin
dy dv du
u v
dx dx dx
d xd x
x x
dx dx
= +
= +

20. 20
Example 6-4. Continuation.
( )2
2
cos sin 2
cos 2 sin
dy
x x x x
dx
x x x x
= +
= +

21. 21
Example 6-5. Determine dy/dx for
the function shown below.
sin x
y
x
=
( ) ( )
2 2
2
sin
sin
cos sin
d x d xdu dv
v u x x
dy dx dx dx dx
dx v x
x x x
x
− −
= =
−
=

22. 22
Example 6-6. Determine dy/dx for
the function shown below.
2
2
x
y e
−
=
2
2
x
u = −
( )
2
2 1
2
2
x
d
du
x x
dx dx
 
− ÷
  = = − = − ÷
 
( )
2 2
2 2
x x
dy
e x xe
dx
− −
= − = −

23. 23
Higher-Order Derivatives
( )y f x=
( )
'( )
dy df x
f x
dx dx
= =
2 2
2 2
( )
''( )
d y d f x d dy
f x
dx dx dx dx
 
= = =  ÷
 
3 3 2
(3)
3 3 2
( )
( )
d y d f x d d y
f x
dx dx dx dx
 
= = =  ÷
 

24. 24
Example 6-7. Determine the 2nd
derivative with respect to x of the
function below.
5sin 4y x=
5(cos4 ) (4 ) 20cos4
dy d
x x x
dx dx
= × =
( )
2
2
20 sin 4 (4 ) 80sin 4
d y d
x x x
dx dx
= − × = −

25. 25
Applications: Maxima and Minima
1. Determine the derivative.
2. Set the derivative to 0 and solve for
values that satisfy the equation.
3. Determine the second derivative.
(a) If second derivative > 0, point is a
minimum.
(b) If second derivative < 0, point is a
maximum.

26. 26
Displacement
Velocity
Acceleration
dy
v
dt
=
Displacement, Velocity, and Acceleration
y
2
2
dv d y
a
dt dt
= =

27. 27
Example 6-8. Determine local maxima
or minima of function below.
3 2
( ) 6 9 2y f x x x x= = − + +
2
3 12 9
dy
x x
dx
= − +
2
3 12 9 0x x− + =
1 and 3x x= =

28. 28
Example 6-8. Continuation.
2
3 12 9
dy
x x
dx
= − +
For x = 1, f”(1) = -6. Point is a maximum and
ymax= 6.
For x = 3, f”(3) = 6. Point is a minimum and
ymin = 2.
2
2
6 12
d y
x
dx
= −