Derivative of a Function

3.1 Derivative of a Function

• Calculus AB

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

Notation

Example 1: Differentiate

Alternate Definition

• Derivative at a point x=a:

• (a, f(a))

• a

Example 3: Use the alternative derivative formula to differentiate at x = a

Example 4: Graph the derivative of the given function.

Example 5: Sketch f given the following information: a.) f(0)=0 b.) The graph below is f’ c.) f is continuous

Vocab

• Right-Hand Derivative:
• Left-Hand Derivative:

Example 6: Show that there are LH & RH derivatives at x = 0 but no derivative

Assignment

• Pg. 105
• 1-18, 21