The document defines the derivative and discusses rules for computing derivatives. It introduces the derivative as describing the slope of a curve at a point. It then outlines several basic rules for determining derivatives, such as the power rule, sum rule, and rules for constants and combinations of functions. The document also discusses the product rule, chain rule, and applications of derivatives to motion and rates of change problems.

1. Calculus 1: Chapter 3
C. A.
September 2012
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3. Definition of the Derivative
 The derivative of a function describes  The slope can be computed using the
the slope of the curve at any point, concept of the limit.
i.e., the slope of a line that is tangent  The process of finding the derivative
to the curve. of a function is called
differentiation.
General definition of the slope of a curve:
Definition of the
Derivative:
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5. Basic rules for derivatives
 Constant rule: The derivative of a constant is zero.
'
f ( x) C f ( x) 0
 Power Rule: n ' n 1
f ( x) x f ( x) nx
d d d
 Sum Rule: ( f ( x) g ( x )) f ( x) g ( x)
dx dx dx
d d
 Multiplication by a constant: ( af ( x )) a f ( x)
dx dx
d d d
 Linear Combination: dx
( af ( x ) bg ( x )) a
dx
f ( x) b
dx
g ( x)
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6. Practice #1
 Find the derivative of each function:
1 1 0
 A. f ( x ) 3 x f ( x) 3x f ( x) 3 1x 3 1 3
Click for Solution
2 1 1
 B. f ( x) 5x f ( x) 5 2x 10 x
Click for Solution
4 1 3 0 3
 C. f ( x ) 7x 5x 2 f ( x) 7 4x 5 1x 0 28 x 5
Click for Solution
 D. 6 Click for Solution
f ( x) 3
x
3 1 4 4 18
f ( x) 6 x f ( x) 6 ( 3) x 18 x 4
x
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7. More rules for derivatives
 Product Rule: d d d
( f ( x ) g ( x )) g ( x) f (x) f ( x) g (x)
 Ex. dx dx dx
2 d
f ( x) (x 1)( 3 x 5 x) ( f ( x ) g ( x )) f '( x) g ( x) f (x)g '( x)
dx
d
( uv ) u'v uv '
dx
 Chain Rule: d
( g ( f ( x ))) g ' ( f ( x )) f ' ( x )
 Ex. dx
2
g ( x) (x 2) dy dy du
u f ( x ), y g (u )
dx du dx
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8. Practice #2
 Find the derivative of each function:
 E. f ( x ) ( 3 x 2 4 x )(1 6 x )
Click for Solution
2
u 3x 4 x, v 1 6x
2
f '(x) (3 x 4 x )( 6 ) (6 x 4 )( 1 6 x)
2 2
( 18 x 24 x ) (6 x 36 x 4 24 x )
2
54 x 42 x 4
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9. Practice #3
 Find the derivative of each function:
 F. f ( x ) ( 7 x 3 2 ) 5
Click for Solution
3 5
u (7 x 2 ), v u
dy 4 3 4 du 2
5u 5(7 x 2) , 21 x
du dx
dy du 3 4 2 2 3 4
f '( x) 5(7 x 2) 21 x 105 x ( 7 x 2)
du dx
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10. Applications of derivatives
 Motion and Rates of change
 Position can be expressed as a
function of time.
 Velocity is the rate of change of
position and can be expressed as
the first derivative of the
position. position x (t )
 Acceleration is the rate of
change of velocity and can be dx
velocity
expressed as the first derivative dt
of the velocity, or the second 2
derivative of the position. dv d x
accelerati on 2
dt dt
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11. Summary
 Derivatives relate to the slope of a function at a point.
The process is called differentiation.
 The derivative of a function can be computed using
limits or a set of rules.
 Derivative of a Constant  Linear Combination
 Power Rule  Product Rule
 Sum Rule  Chain Rule
 Multiplication by a Constant
 Derivatives are applicable for many word problems
involving rates of change.
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12. Additional Resources
 http://en.wikipedia.org/wiki/Derivative_(calculus)
 http://archives.math.utk.edu/visual.calculus/2/definit
ion.8/index.html
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