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So lets look at an example:
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Find the extrema of
on the interval [-1, 2].
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Implicit Differentiation
When it is not easy (or possible) to isolate y from the rest of the equation, we can
determine the derivative using implicit differentiation.
Consider the circle x2 + y2 = 25. We will find its derivative using two methods:
1. Isol
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Sign Chart Practice
Given the following derivative and critical values, create the sign chart.
1.
f ' ( x) 6 x 2 28 x 22
0 2( x 1)(3x 11)
x 1, x 113
2.
f ' ( x) 4 x 3 12 x 2
0 4 x 2 ( x 3)
x 0, x 3
3.
f ' ( x)
1 x2
x(1 x 2 )
0 1 x2
x 1, x 1
0 x(1 x 2 )
x
Name:
Class Section (A or B)
Worksheet: Chapter 8 Section 1: Higher Order Derivatives
1. Find the first, second, and third derivatives for y = 3x4 2x3 + 7x2 + 9x 12.
2. Find the first, second, and third derivatives for y = 2 ln x.
3. Find the first, secon
Worksheet: Chapter 8 Section 1: Higher Order Derivatives: ANSWERS
1. Find the first, second, and third derivatives for y = 3x4 2x3 + 7x2 + 9x 12.
y 0 = 12x3 6x2 + 14x + 9
y 00 = 36x2 12x + 14
y 000 = 72x 12
2. Find the first, second, and third derivatives
The Chain Rule
The derivative of a function f ( x) g (h( x) can be determined using the
chain rule:
f ( x) g (h( x)h( x)
or
dy dy du
dx du dx
For example, the function f ( x) x 2 1 is composed of g ( x ) x and
h( x ) x 2 1 . To calculate the derivative, a
Trigonometric Functions
Limits as 0 of trigonometric functions:
lim sin
lim cos
0
0
lim tan
0
Fundamental Trigonometric Limit:
lim
0
sin 7
.
0
4
Determine lim
What is lim cot ?
0
Calculate lim
0
cos 1
.
sin
1
Derivatives of trigonometric functi
Exponential and Logarithmic Functions
The formula for compound interest is:
r
A P 1
n
nt
As n increases, the number of compounding intervals increases. Is there a limit to the number
of compounding periods possible per year?
Let h r . As n increases, t
Differentiation Rules
1. The derivative of a constant is 0:
If f(x) = c then f'(x) = 0.
(Think of the slope of a horizontal line.)
2. If f(x) = cx then f'(x) = c.
3. If f(x) = xn then f'(x) = nxn - 1.
The derivative of a function made up of a sum or diffe