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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.