chapter 4 calculating the derivative
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Chapter 4 Calculating the Derivative. JMerrill, 2009. Review. Find the derivative of (3x – 2x 2 )(5 + 4x) -24x 2 + 4x + 15 Find the derivative of. 4.3 The Chain Rule. Composition of Functions. A composition of functions is simply putting 2 functions together—one inside the other. - PowerPoint PPT PresentationTRANSCRIPT
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Chapter 4Calculating the
Derivative
JMerrill, 2009
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Review
Find the derivative of (3x – 2x2)(5 + 4x)
-24x2 + 4x + 15 Find the derivative of
25x 2x 1
22 2
5x 4x 5(x 1)
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4.3The Chain Rule
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Composition of Functions
A composition of functions is simply putting 2 functions together—one inside the other.
Example: In order to convert Fahrenheit to Kelvin we have to use a 2-step process by first converting Fahrenheit to Celsius.
89oF = 31.7oC 31.7oC = 304.7K But if we put 1 function inside the other function,
then it is a 1-step process.
5C (F 32) 9K C 273
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Composition of Functions
We are used to writing f(x). f(g(x)) simply means that g(x) is our new x in the f equation.
We can also go the other way. means g(f(x)).
The composite of f(x) and g(x) is denoted which means the same as f(g(x)).
f g x
g f x
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Given 2( ) 4 2 ( ) 2f x x x g x x
f(g(3)) =
= f(6)
= 4(6)2 – 2(6)
= 144 – 12
= 132
g(3) = 6
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Given 1( ) ( ) 1f x g x xx
f g x
( ( ))f g x ( ( ))( 1)f g xf x
( ( ))( 1)1
1
f g xf x
x
g f x
( ( ))1
1 1
g f x
gx
x
g(x) = x+1
Substitute x+1In place of the
x in the f equation
=
The new x in the g equation
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The Chain Rule
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Chain Rule Example
Use the chain rule to find Dx(x2 + 5x)8
Let u = x2 + 5x Let y = u8
72
7
dy dy dudx du dx 8u 2
8 x 5x
x 5
2x 5
Another way to think of it: The derivative of the outside times the derivative of the inside
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Chain Rule – You Try
Use the chain rule to find Dx(3x - 2x2)3
Let u = 3x - 2x2 Let y = u3
22
2
dy dy dudx du dx 3u 3 4x
3x 2x 3 4x3
The derivative of the outside times the derivative of the inside
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Chain Rule
Find the derivative of y = 4x(3x + 5)5
This is the Product Rule inside the Chain Rule. Let u = 3x + 5; y = u5
4
4 5
4 5
4 5
5
4x 5u (3) (3x 5) (4)
4x 5(3x 5) (3) 4(3x 5)
4x 15(3x 5) 4(360x(3x 5)
x 5)4(3x 5)
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Chain Rule
4
4
4
5
Factor out the common f actor14(3x 5)
60x(3x 5) 4(3x 5)
4(3x 5) (18x 55x (3x 5)
)
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Chain Rule
Find the derivative of This is the Quotient Rule in the Chain Rule Let u = 3x + 2; let y = u7
73x 2x 1
6 7
2
6
2
7
2
6 7
(x 1) 7u (3) (3x 2) (1)(x 1)
(x 1) 7(
21 (x 1)(3x 2) (3x 2
3x 2) (3) (3x 2)(x 1)
)(x 1)
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Chain Rule
6
6 7
2
6
2
2
2
6
Factor out the common f actor21(x 1) (3x 2)(x
(3x 2)
(
21 (x 1)(3x 2) (3x 2)(x 1)
(3x 2)
3
1
x
8
1)21x 21 3x 2(x 1)
x 2)
2
3
)
(x 1
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4.4 Derivatives of Exponential Functions
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Derivative of ex
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Derivative of ax
xx
x(lD 3 n3)3
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Other Derivatives
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Examples – Find the Derivative
y = e5x
g(x)
x 5x5e (g'(x)e (5) 5e
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Examples – Find the Derivative
y = 32x+1
g(x)
2
2x 1
x 1
lna a g'(x)
ln3 32ln3 3
(2)
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Example
Find if Use the product rule
1
21 5x 2 (5)2
52 5x 2
2x 1y e 5x 2 dydx
12 2x 1 x 12x xy e D 5x 2 5x 2 D e
2x 1e (2x)
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Example
12 2x 1 x 12x x
2 2x 1 x 1
2x 1 2x 1
y e D 5x 2 5x 2 D e
5e 5x 2 2xe2 5x 25e 5x 2 2xe2 5x 2
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Example Continued
2x 1 2x 1
2x 1 2
2 2x 1 x 1
2x 1
5e 2xe 5x 22 5x 2
5e e (4x)(5x 2)2 5x
e 2
2e 5 4x(5x 2)
2 5x
2 5x 2
0x 8x 52 5x
2 5 2
2
x
2
Get a common denominator to add the 2 parts together
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4.5Derivatives of Logarithmic Functions
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Definition
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Bases – a side note Everything we do is in Base 10.
We count up to 9, then start over. We change our numbering every 10 units. 1 11 212 12 223 13 23…4 145 156 167 178 189 1910 20
Ones Place
One group of ten and 1, 2, 3…
ones
Two tens
and …ones
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Bases The Yuki of Northern California used Base 8.
They counted up to 7, then started over. The numbering changed every 8 units.
1 13 252 14 263 15 27…4 165 176 207 2110 2211 2312 24
Ones Place
One eight
and 3…ones
Two eights and …ones
So, 17 in Base 8 = 15 in Base 10
258 = 2 eights + 5 ones = 21
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Bases
The Mayans used Base 20. The Sumerians and people of Mesopotamia
used Base 60.
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Definition
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Example
Find f’(x) if f(x) = ln 6x Remember the properties of logs ln 6x = ln 6 + ln x
d d(ln6) (lnx)dx dx10 1
xx
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Definitions
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Examples – Find the Derivatives
y = ln 5x If g(x) = 5x, then g’(x) = 5
dy g'(x) 5dx g(x) 5
1xx
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F’(x)
f(x) = 3x ln x2
Product Rule
2 2
2
2
2
df ' (x) (3x) lnx lnx (3)
6 3lnx
dx2x3x lnx (3)x