calculus 45s slides march 17, 2008

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

Proof:

If then

The Derivative of a Constant Function

Proof:

If then

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: