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.
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1 Chapter 6 Differential Calculus The two basic forms of calculus are differential calculus and integral calculus. This chapter will be devoted to the.
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1
Chapter 6Differential 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.
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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.
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The derivative of a function is the slope at a given point.
x
y
( )y f x
x
y
4
Various Symbols for the Derivative
( ) or '( ) or
dy df xf x
dx dx
0Definition: lim
x
dy y
dx x
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Figure 6-2(a). Piecewise Linear Function (Continuous).
(a) ( )y f x
x
Continuous Function
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Figure 6-2(b). Piecewise Linear Function (Finite Discontinuities).
(b)
Discontinuities
( )y f x
x1x 2x 3x
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Piecewise Linear Segment
1 1( , )x y
2 2( , )x y
2 1y y
2 1x x
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Slope of a Piecewise Linear Segment
2 1
2 1
slopey ydy
dx x x
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Example 6-1. Plot the first derivative of the function shown below.
2 4 6 8 10 x
( )y f x12
-12
(a)
10
x
'( )dy
f xdx
3
-12
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2 4 6 8 10 x
( )y f x12
-12
(a)
(b)
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Development of a Simple Derivative
2y x2( )y y x x
2 22 ( )y y x x x x
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Development of a Simple DerivativeContinuation
22 ( )y x x x
2y
x xx
0lim 2x
dy yx
dx x
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Chain Rule
( )y f u ( )u u x( )
'( )dy df u du du
f udx du dx dx
( )'( )
df uf u
duwhere
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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
2.010.01
dy y
dx x
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Example 6-3. The derivative of sin u with respect to u is given below.
sin cosd
u udu
Use the chain rule to find the derivative with respect to x of
24siny x
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Example 6-3. Continuation.
2u x
2du
xdx
2
'( )
4(cos )(2 ) 8 cos
dy du dy duf u
dx dx du dx
u x x x
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Table 6-1. Derivatives( )f x '( )f x Derivative Number
( )af x '( )af x D-1
( ) ( )ux vx '( ) '( )u x v x D-2
( )f u ( )'( )du df u du
f udx du dx
D-3
a 0 D-4
( 0)nx n 1nnx D-5
( 0)nu n 1n dunudx
D-6
uv dv duu vdx dx D-7
u
v2
du dvv udx dxv
D-8
ue uduedx
D-9
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Table 6-1. Derivatives (Continued)ua l n u d u
a ad x
D - 1 0
l n u 1 d u
u d xD - 1 1
l o g a u 1l o g a
d ue
u d xD - 1 2
s i n uc o s
d uu
d x
D - 1 3
c o s us i n
d uu
d x D - 1 4
t a n u 2s e cd u
ud x
D - 1 5
1s i n u1
2
1 s i n
2 21
d uu
d xu
D - 1 6
1c o s u
2
1
1
d u
d xu
1 0 c o s u D - 1 7
1t a n u1
2
1 t a n
1 2 2
d uu
u d x
D - 1 8
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Example 6-4. Determine dy/dx for the function shown below.
2 siny x x
2
2 sinsin
dy dv duu v
dx dx dx
d xd xx x
dx dx
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Example 6-4. Continuation.
2
2
cos sin 2
cos 2 sin
dyx x x x
dx
x x x x
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Example 6-5. Determine dy/dx for the function shown below.
sin xy
x
2 2
2
sinsin
cos sin
d x d xdu dvv u x xdy dx dx dx dx
dx v xx x x
x
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Example 6-6. Determine dy/dx for the function shown below.
2
2
x
y e
2
2
xu
2
2 12
2
xd
dux x
dx dx
2 2
2 2
x xdye x xe
dx
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Higher-Order Derivatives
( )y f x( )
'( )dy df x
f xdx dx
2 2
2 2
( )''( )
d y d f x d dyf x
dx dx dx dx
3 3 2(3)
3 3 2
( )( )
d y d f x d d yf x
dx dx dx dx
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Example 6-7. Determine the 2nd derivative with respect to x of the function below.
5sin 4y x
5(cos 4 ) (4 ) 20cos 4dy d
x x xdx dx
2
220 sin 4 (4 ) 80sin 4
d y dx x x
dx dx
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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.
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Displacement
Velocity
Acceleration
dyv
dt
Displacement, Velocity, and Acceleration
y
2
2
dv d ya
dt dt
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Example 6-8. Determine local maxima or minima of function below.
3 2( ) 6 9 2y f x x x x 23 12 9
dyx x
dx
23 12 9 0x x
1 and 3x x
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Example 6-8. Continuation.
23 12 9dy
x xdx
For x = 1, f”(1) = -6. Point is a maximum and ymax= 6.
For x = 3, f”(3) = 6. Point is a minimum andymin = 2.