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Here we will be discussing the basics of a derivative of functions.

1.
3.1 Derivative of a

Function

Calculus AB

Function

Calculus AB

2.
Vocab/Formulas

• Derivative:

– From 2.4 we defined slope of a curve of y = f(x) at

a point where x=a as

– When it exists, this limit is called the derivative

of f at a. f’(x) is the derivative of the function.

• Differentiable: a function is differentiable

at a point when the f’(x) exists

– If the function is differentiable at every point in

the domain, it is a differentiable function

• Derivative:

– From 2.4 we defined slope of a curve of y = f(x) at

a point where x=a as

– When it exists, this limit is called the derivative

of f at a. f’(x) is the derivative of the function.

• Differentiable: a function is differentiable

at a point when the f’(x) exists

– If the function is differentiable at every point in

the domain, it is a differentiable function

3.

4.

5.
Example 1: Differentiate

6.

7.
Alternate Definition

• Derivative at a point x=a:

(a,

f(a))

a

• Derivative at a point x=a:

(a,

f(a))

a

8.
Example 3: Use the alternative

derivative formula to differentiate at x = a

derivative formula to differentiate at x = a

9.
Example 4: Graph the derivative of

the given function.

the given function.

10.
Example 5: Sketch f given the

following information:

a.) f(0)=0 b.) The graph below is f’ c.) f is

continuous

following information:

a.) f(0)=0 b.) The graph below is f’ c.) f is

continuous

11.
Vocab

• Right-Hand Derivative:

• Left-Hand Derivative:

• Right-Hand Derivative:

• Left-Hand Derivative:

12.
Example 6: Show that there are LH

& RH derivatives at x = 0 but no derivative

& RH derivatives at x = 0 but no derivative

13.
Pg. 105

1-18, 21

1-18, 21