Chapter 3.1
The Derivative
3.1.1 The Tangent Question and the Derivatives
Average Rate of Change at a Point
from (
The average rate of change of the function
(
)
(
(
)
( )) to
)) is the ratio
(
where
(
)
( )
.
Slope of a Tangent to a Graph
The slope of a tangent to the graph of
( ) at the point (
(
)
provided that this limit exists.
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( )
( )) is
Derivative of a Function
at a Number
If
( ) defines a function and if the point is in the domain of , then the derivative of
at , written as ( ), is defines as
(
( )
)
( )
provided that this limit exists.
Derivative of a Function
The derivative of a function
is the function
defined as
(
( )
)
( )
if the limit exists.
Note: The notation
,
( ),
,
( ) is not a unique notation for the derivative. Other notations include
( ),
( ) . There are times when a meaning is clearer if one of
, and
these notations is used instead of
for a derivative. The symbol
( ). The symbols
and
are not fractions but are symbols
indicates that we are going to find a derivative.
Four-Step Method
We can calculate the derivative of a function
Four Steps for Calculating
Step 1: Find (
at the point by using the following four steps.
( )
).
Step 2: Subtract ( ) from (
Step 3: Divide the result in Step 2 by
).
to get the difference quotient.
Step 4: Find the limit, if it exists, of the difference quotient in Step 3, as
Example 1:
Find the slope of ( )
,
at the point (
).
2
approaches .
3.1.2 Derivatives of Polynomials
Constant Function
For the function ( )
, where
( )
is a constant, the derivative is
.
Linear Function
For the linear function ( )
, the derivative is
( )
.
Polynomials of the Form ( )
The derivative of any function ( )
, if
is a rational number, then
Example 2
Find the derivative of ( )
at
.
3
( )
.
Example 3:
Find the derivative of ( )
.
Example 4:
Find the derivative of (a) ( )
, (b) ( )
, (c) ( )
⁄
, (d) ( )
Example 5:
Find the derivatives of (a)
√ and (b) ( )
⁄
.
Three General Formulae
If ( )
( ),
( ) exists, then
is a constant and
( )
If ( )
( )
( ) and
( ) and
( ) exist, then
( )
If ( )
( )
( ) and
( ) and
( )
( )
( )
( ) exist, then
( )
( )
( )
Example 6:
Find the derivative of (a) ( )
, (b) ( )
, and (c) ( )
Example 7:
Find the derivative of ( )
.
4
⁄
.
⁄
.
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3.1.3 Derivatives of Product and Quotients
Product Rule for Derivatives
If ( )
( ) ( ) and
( ) and
( ) exist, then
( )
( ) ( )
Example 8:
Find the derivative of ( )
(
)(
).
Example 9:
Find
( ), if ( )
(
)(
).
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( ) ( )
Quotient Rule for Derivatives
If ( )
( )
( )
, ( )
and both
( ) and
( )
( ) exist, then
( ) ( )
( ) ( )
( )
Example 10:
If ( )
, find
( ).
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3.1.4 Derivatives of Composite Functions
General Power Rule for Derivatives
If
is a function and its derivative
exists, then
( )
( )
( )
Example 11:
Find the derivatives of (a) (
) , (b) (
) , and (c) (
8
) .
Chain Rule
If
( ) and
( ), and both
( ) ( )
( ) and
( ) exist, then
( )
or
Example 12:
Find
, if (a)
(
)
√
and (b)
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( )
3.1.5 Implicit Differentiation
Four Steps for Implicit Differentiation
In the list below, we assume that
determine
is the independent variable and that we are trying to
.
1. Differentiate both sides of the equation with respect to the independent variable, .
2. Collect the terms with
on one side of the equation and collect the remaining terms on
the other side of the equals sign.
3. Factor out
4. Solve for
.
.
Example 13:
Find
of
.
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Example 14:
Use implicit differentiation to find the derivative of
.
3.1.6 Higher Order Derivatives
The derivative of a function ( ), is also the function ( ), called the derivative function .
When we need to take the derivative of the derivative is called the second derivative and is
denoted by . The derivative of the second derivative is the third derivative, .
In general, if is a positive integer, then ( ) represents the th derivative of . The integer is
called the order of the derivative ( ) and, as a group, these are known as higher order
derivatives.
Example 15:
Find the first four derivatives of ( )
.
Example 16:
Find the first three derivatives of ( )
.
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