slide 3- 1 what you’ll learn about definition of a derivative notation relationship between the...
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Slide 3- 1
What you’ll learn about• Definition of a Derivative• Notation• Relationship between the Graphs of f and f '• Graphing the Derivative from Data• One-sided Derivatives
… and why
The derivative gives the value of the slope of the tangent line to a
curve at a point.
Slide 3- 2
Definition of Derivative
0
The of the function with respect to the variable is the
function whose value at is
lim
provided the limit exists.
h
f x
f x x
f x h f xf x
h
derivative
Slide 3- 3
Example Definition of Derivative
2Differentiate f x x
0
2 2
substitute0
2 2 2
2expanded
0
2cancelled and h factored out
0
0
Applying the definition, we have
lim
lim
2lim
2lim
= lim 2
2
h
h
x hh
xh
h
f x h f xf x
h
x h x
h
x xh h x
hx h h
hx h
x
Slide 3- 4
Derivative at a Point (alternate)
The derivative of the function at the point where is the limit
lim
provided the limit exists.
x a
f x a
f x f af a
x a
Slide 3- 5
Differentiable Function
The domain of , the set of points in the domain of for which the limit
exists, may be smaller than the domain of . If exists, we say that
at . A
f f
f f x f
x
has a derivative (is differentiable) function that is differentiable
at every point in its domain is a .differentiable function
0
limh
f a h f a
h
is called the derivative of at .f a
We write: 0
limh
f x h f xf x
h
“The derivative of f with respect to x is …”
There are many ways to write the derivative of y f x
f x “f prime x” or “the derivative of f with respect to x”
y “y prime”
dy
dx“dee why dee ecks” or “the derivative of y with
respect to x”
df
dx“dee eff dee ecks” or “the derivative of f with
respect to x”
df x
dx“dee dee ecks uv eff uv ecks” or “the derivative
of f of x”( of of )d dx f x
dx does not mean d times x !
dy does not mean d times y !
dy
dx does not mean !dy dx
(except when it is convenient to think of it as division.)
df
dxdoes not mean !df dx
(except when it is convenient to think of it as division.)
(except when it is convenient to treat it that way.)
df x
dxdoes not mean times !
d
dx f x
y f x
y f x
The derivative is the slope of the original function.
The derivative is defined at the end points of a function on a closed interval.
2 3y x
2 2
0
3 3limh
x h xy
h
2 2 2
0
2limh
x xh h xy
h
2y x
0lim 2h
y x h
0
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.
p
Slide 3- 14
Example Definition of Derivative
2Differentiate f x x
0
2 2
substitute0
2 2 2
2expanded
0
2cancelled and h factored out
0
0
Applying the definition, we have
lim
lim
2lim
2lim
= lim 2
2
h
h
x hh
xh
h
f x h f xf x
h
x h x
h
x xh h x
hx h h
hx h
x
Slide 3- 15
Example One-sided DerivativesShow that the following function has left-hand and right-hand
derivatives at 0, but no derivative there.
, 0
, 0
x
x xy
x x
0 0
0 0
Left-hand derivative: Right-hand derivative:
0 0 0 0lim lim
lim 1 lim 1
The derivatives are not equal at 0. The function does not
have a derivative at 0.
h h
h h
h h
h hh h
h h
x