The Definition of a Derivative




   Mathematics is the science of pattern ... let's look for some patterns ...


    Let...
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 t...
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:
Calculus 45S Slides March 17, 2008
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Calculus 45S Slides March 17, 2008

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Proofs of the power rule, constant multiplier rule, sum and difference rules.

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Calculus 45S Slides March 17, 2008

  1. 1. 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 ...
  2. 2. The Derivative of a Linear Function If then Proof:
  3. 3. The Derivative of a Constant Function If then Proof: Example: Find the derivative of
  4. 4. The Derivative of a Power Function Proof:
  5. 5. Expand and simplify ... a+b
  6. 6. 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
  7. 7. The Binomial Theorem ...
  8. 8. The Derivative of a Power Function Proof:
  9. 9. The Derivative of a Constant Times a Function Proof: Example: Find the derivative of
  10. 10. The Derivative of a Sum of Functions Proof:
  11. 11. The Derivative of a Difference of Functions Proof:
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