3.1 derivatives great sand dunes national monument, colorado photo by vickie kelly, 2003 created by...

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.