COMP 206 Mathematics

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COMP 206 Mathematics

• Instructor Zhijun Wang
• Partial of homework 1 is available online now
• Todays Content
• Rules for differentiation
• Derivative of exponential and logarithm functions
• Derivative of trigonometric functions

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Some rules of differentiation

• Three traditional rules greatly extend the number
  of functions that we can differentiate

1. Constant-multiple rule
k is a constant
2. Sum rule
3. General power rule
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Application of constant-multiple rules
Constant-multiple rule
k is a constant
Calculate
(b)
(a)
(c)
Answers
(a) k2 and
So we have
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Application of constant-multiple rules
Answers
So we have
(b) k1/4, and
(c) Write
Then
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Application of sum rule
sum rule
Calculate
(a)
(c)
(b)
Answers
and
(a) let
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Application of sum rule
Answers
and
(b) let
(c) We apply the sum rule repeatedly, we have
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Application of general power rule
General power rule
Calculate
(a)
(b)
Answers
(a)
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Application of general power rule
Answers
(b)
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Variables of Derivative
In some applications, we may use parameters other
than x and y, for example, f(t)2t. The
derivative of f(t) corresponds to t can be
written as
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Examples
Compute
if
(a)
(b)
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Answers
(a)
(b)
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The second derivative
When we differentiate a function f(x), we obtain
another function f(x) that is a formula for the
slope of the curve yf(x). If we differentiate
the function f(x), we obtain the second
derivative of f(x), denoted by f(x). That is
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Other notations of second derivative
If y is a function of x say yf(x), we have the
following notations
Prime Notation
Notation
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Examples
1. Compute the second derivative of the following
functions
(a)
(b)
, find
2. If
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Answers
1. Compute the second derivative of the following
functions
(a)
(b)
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Answers
2. If
, find
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Questions??