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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 ...
The Derivative of a Linear Function

         If                then

Proof:
The Derivative of a Constant Function

         If                 then

Proof:




 Example: Find the derivative of
The Derivative of a Power Function




Proof:
Expand and simplify ...



           a+b
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
The Binomial Theorem ...
The Derivative of a Power Function




Proof:
The Derivative of a Constant Times a Function




Proof:




Example: Find the derivative of
The Derivative of a Sum of Functions




Proof:
The Derivative of a Difference of Functions




Proof:

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