Derivatives Great Sand Dunes National Monument, Colorado Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003

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 ! (Well does not mean ! (except when it is convenient to think of it as division.)... Sometimes we use it that way)

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

In the future, all will become clear.

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.  Differentiability implies Continuity, but Continuity does not imply Differentiability

To be differentiable, a function must be continuous and smooth. Derivatives will fail to exist at: cornercusp vertical tangent discontinuity

Most of the functions we study in calculus will be differentiable.

If f has a derivative at x = a, then f is continuous at x = a. Since a function must be continuous to have a derivative, if it has a derivative then it is continuous.

Intermediate Value Theorem for Derivatives Between a and b, must take on every value between and. If a and b are any two points in an interval on which f is differentiable, then takes on every value between and. 