3.1 Derivatives Part 2.

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Presentation transcript:

3.1 Derivatives Part 2

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

  The Definition of the Derivative: y2 y1 x2 x1 The Derivative at a Point:  x2 x1 There are many ways to write the derivative of

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

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

Note: does not mean ! 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.)

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

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

The derivative is the slope of the original function. f (x) f ´(x)

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. p