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COMP 206 Mathematics

• Instructor Zhijun Wang
• Todays Content
• Rules for differentiation
• Derivative of exponential and logarithm functions
• Derivative of trigonometric functions

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Derivative of exponential function

• 1.

e2.718.
Proof
We can show
So
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Derivative of exponential function

• 2.

Examples
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Chain rule for exponential functions

• Chain rule

Or let ug(x), then we have
If g(x)kx, we have
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Examples

• Calculate the derivatives of the following
  functions

(a)
(b)
Answers
(a)
(b)
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Review of logarithm functions

• Remember that

Let x and y be positive numbers, b any number, we
have
(a)
(b)
(c)
Note that ln10 and loga10
(d)
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Derivative of logarithm functions

• 1.

Proof Since
, we have
In the other hand, using chain rule, we have
i.e.
or
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Chain rule of logarithm functions

• Chain rule

If ug(x), we can write it as
General logarithm function
Note that
Let be
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Examples

• Calculate

(a)
(b)
Answers
(a)
(b)
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Excises

• Calculate

(a)
(b)
(c)
(d)
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Answers
(a)
(b)
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Answers
(c)
(d)
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Derivative of sine and cosine
We can use the limit definition to prove them.
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Chain rule
Calculate
(a)
(b)
(c)
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Answers
Answers
(a)
(b)
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Answers
Answers
(c)
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General Power rule
Composition function assume f(x) and g(x) are
two functions, then f(g(x)) is called a
composition (or composite) function.
Example
What is f(g(x))?
Ans Replace each x in f(x) by g(x) to obtain
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General Chain Rule

• General Chain rule To differentiate f(g(x)),
  first differentiate the outside function f(x) and
  substitute g(x) for x in the result. Then
  multiple by the derivative of inside function
  g(x). Symbolically,

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Application of general chain rule

• Example 1

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Application of general chain rule

• Example 2 Find

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How to get these derivatives?
(a)
(b)
(c)
(d)
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Product rule
Calculate
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Application
During a rocket launching, the ground station
measures the distance of the rocket is a function
of time t as
What is the rocket speed at time t10.
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Solution
The speed of the rocket at time t is the distance
change rate at time t, the change rate is the
derivate of d(t). So
So v(10)50
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Excises
Calculate
(a)
(b)
(c)
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Answers
(a)
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Answers
(b)
(c)
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Quotient rule
Calculate
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Excises
Calculate
(b)
(a)
(c)
Answer (a)
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Answers
(b)
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Answers
(c)
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Derivative of xx
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Summary of derivatives
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Summary of chain rule
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Summary of derivative rules
Const-multiple Sum rule General power
rule Product rule Quotient rule
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L'Hopital's Rule
If
Example
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Questions??
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Questions??