Download - Calculus 45S Slides March 17, 2008
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The Definition of a Derivative
Mathematics is the science of pattern ... let's look for some patterns ...
Let's use the definition to find some derivatives ...
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The Derivative of a Linear Function
Proof:
If then
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The Derivative of a Constant Function
Proof:
If then
Example: Find the derivative of
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The Derivative of a Power Function
Proof:
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Expand and simplify ...
a + b
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The Binomial Theorem ...
Algebraically
Combinatorically
Notice the patterns ...
(1) The coefficient of the term is:
(2) The exponent on a is given by: [n - (i - 1)]
(3) The exponent on b is given by: i - 1
(4) This relation holds for each term in the expansion: [exponent on a] + [exponent on b] = n(5) The number of terms in any binomial expansion is: n + 1
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The Binomial Theorem ...
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The Derivative of a Power Function
Proof:
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The Derivative of a Constant Times a Function
Proof:
Example: Find the derivative of
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The Derivative of a Sum of Functions
Proof:
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The Derivative of a Difference of Functions
Proof: