3.1 DerivativesGreat Sand Dunes National Monument, Colorado

Photo by Vickie Kelly, 2003Created by Greg Kelly, Hanford High School, Richland, WashingtonRevised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts

0

limh

af fh a

h

is called the derivative of at .f a

We write: 0

' limh

ff a

af a

h

h

“The derivative of f at a is the limit of the difference quotientas h approaches 0…”

There are many ways to write and to verbalizederivatives …

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 …”

( ) ( )( ) lim

) )'

( (x a

x aa

xf

f f

a

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 x “f prime of x” or “the derivative of f with respect to x”

'y “y prime”

dy

dx“dee y dee x” or “the derivative of y

with respect to x”

“dee f dee x” or “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”

or “the derivative of y ”

df

dx

( )d

df

xx

dx does not mean d times x !

dy does not mean d times y !

df x

dxdoes not mean times !

d

dx f x

dy

dx does not mean !dy dx

df

dxdoes not mean !df dx

In the future, all will become clearer...

y f x

y f x

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!)

2 3y x

2 2

0

( 3) 3limh

x h xy

h

2 2 2

0

2 3 3limh

x xh h xy

h

2y x

0lim 2h

y x h

0

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.