Slide 3- 1 What you’ll learn about Definition of a Derivative Notation Relationship between the Graphs of f and f ' Graphing the Derivative from Data One-sided.

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Presentation transcript:

Slide 3- 1 What you’ll learn about Definition of a Derivative Notation Relationship between the Graphs of f and f ' Graphing the Derivative from Data One-sided Derivatives … and why The derivative gives the value of the slope of the tangent line to a curve at a point.

Slide 3- 2 Definition of Derivative

Slide 3- 3 Example Definition of Derivative

Slide 3- 4 Derivative at a Point (alternate)

Slide 3- 5 Differentiable Function

is called the derivative of at. We write: “The derivative of f with respect to x is …” There are many ways to write the derivative of

“f prime x”or “the derivative of f with respect to x” “y prime” “dee why dee ecks” or “the derivative of y with respect to x” “dee eff dee ecks” or “the derivative of f with respect to x” “dee dee ecks uv eff uv ecks”or “the derivative of f of x”

dx does not mean d times x ! dy does not mean d times y !

does not mean ! (except when it is convenient to think of it as division.) does not mean ! (except when it is convenient to think of it as division.)

(except when it is convenient to treat it that way.) does not mean times !

The derivative is the slope of the original function. The derivative is defined at the end points of a function on a closed interval.

A function is differentiable if it has a derivative everywhere in its domain. It must be continuous and smooth. Functions on closed intervals must have one-sided derivatives defined at the end points. 

Slide Example Definition of Derivative

Slide Example One-sided Derivatives