3.1 Derivatives Great Sand Dunes National Monument, Colorado Photo by Vickie Kelly, 2003 Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts

is called the derivative of at. We write: The derivative of f at a is the limit of the difference quotient as h approaches 0… There are many ways to write and to verbalize derivatives …

One alternative definition of the derivative of f with respect to x at the point x = a …(handy for radical functions or fractions) The derivative of f (with respect to x) at a is …

There are many traditions for writing derivatives -- some from Newton, some from Leibniz, some from function notation -- with subtle differences in emphasis, but the same overall meaning!

f prime of x or the derivative of f with respect to x y prime dee y dee xor the derivative of y with respect to x dee f dee xor the derivative of f with respect to x dee dee x of f of x or the derivative of f of x with respect to x orthe derivative of y

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

does not mean !

In the future, all will become clearer...

The derivative of f is the slope along f. The one-sided derivative is defined at the end point (original f has a closed interval!)

A function is differentiable if it has a derivative everywhere (at every point!) in its domain. At each end point on closed intervals, continuous functions have one-sided derivatives defined. Differentiable regions are continuous and locally linear when you zoom in on them.