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Limit Definition of the
Derivative
Objective
To use the limit definition to find the
derivative of a function.
.
What is a derivative?
A function
the rate of change of a
function
the slope of the line tangent to
the curve
The tangent line
single point
of intersection
slope of a secant line
ax
f(x)
f(a)
f(a) - f(x)
a - x
slope of a (closer) secant line
ax
f(x)
f(a)
f(a) - f(x)
a - x
x
closer and closer…
a
watch the slope...
watch what x does...
ax
The slope of the secant line gets closer and closer
to the slope of the tangent line...
As the values of x get closer and closer to a!
ax
The slope of the secant lines
gets closer
to the slope of the tangent line...
...as the values of x
get closer to a
Translates to….
limax
f(x) - f(a)
x - a
Equation for the slope
Which gives us the the exact slope
of the line tangent to the curve at a!
as x goes to a
The Difference Quotient
The derivative is the slope of the tangent
line to a graph f(x), and is usually denoted
f’(x).
To calculate the slope of the tangent line
we will use the difference quotient.
The Difference Quotient
Limit Definition of the Derivative
The derivative is the formula which gives
the slope of the tangent line at any point x
for f (x), and is denoted
provided this limit exists.
0
( ) ( )'( ) lim
x
f x x f xf x
x
Derivatives
The derivative of the function y = f (x) may be
expressed as …
'( )f x
'y
dy
dx
Prime notation
Leibniz notation
“f prime of x”
“y prime”
“the derivative of y with respect to x”
Derivatives
The process of finding derivatives is called
differentiation.
A function is differentiable at a point if its
derivative exists at that point.
Limit Definition of the Derivative
Use the limit definition to find the
derivative of:2( ) 3 5f x x x
0
( ) ( )'( ) lim
x
f x x f xf x
x
Limit Definition of the Derivative
2( ) 3 5f x x x
0
( ) ( )'( ) lim
x
f x x f xf x
x
2 2
0
( ) 3( ) 5 ( 3 5)'( ) lim
x
x x x x x xf x
x
2 2 2
0
2 ( ) 3 3 5 3 5'( ) lim
x
x x x x x x x xf x
x
Limit Definition of the Derivative
0
(2 3)'( ) lim
x
x x xf x
x
2
0
2 ( ) 3'( ) lim
x
x x x xf x
x
'( ) 2 3f x x
A formula for finding the
slope of the tangent line
of f (x) at a given point.
Limit Definition of the Derivative
Use the limit definition to find the
derivative of:2( ) 8 1f x x
0
( ) ( )'( ) lim
x
f x x f xf x
x
Limit Definition of the Derivative
2( ) 8 1f x x
0
( ) ( )'( ) lim
x
f x x f xf x
x
2 2
0
8( ) 1 (8 1)'( ) lim
x
x x xf x
x
2 2 2
0
8( 2 ( ) ) 1 8 1'( ) lim
x
x x x x xf x
x
Limit Definition of the Derivative
0
(16 8 )'( ) lim
x
x x xf x
x
'( ) 16f x x
A formula for finding the
slope of the tangent line
of f (x) at a given point.
2 2 2
0
8 16 8( ) 1 8 1'( ) lim
x
x x x x xf x
x
similarly...
aa+h
f(a+h)
f(a)
f(x+h) - f(x)
(x+h) - x
= f(x+h) - f(x)
h
(For this particular curve, h is a negative value)
h
thus...
lim f(a+h) - f(a)
hh 0
AND
lim f(x) - f(a)x a x - a
Give us a way to calculate the slope of the line tangent at a!
Which one should I use?
(doesn’t really matter)
A VERY simple example...
want the slope
where a=2
xy 2xy
2xy
ax
axax
ax
ax
ax
afxf
))((limlim
)()(lim
22
4)2lim()lim( xax
as x a=2
h
xhx
h
xfhxf 22)(lim
)()(lim
h
hxh
h
xhxhx )2(lim
2lim
222
4)2lim( hx
As h 0
back to our example...
When a=2,
the slope is 4
xy 2xy
2xy
Differentiability
Not every function is differentiable at all points.
Some common situations in which a function will not be differentiable at a point include:
1. Vertical tangent lines
2. Discontinuities (like a hole, break, or vertical
asymptote)
3. Sharp turns (called cusps & nodes)
Differentiability
Differentiability
Differentiability
Differentiability
CALCULUS JEOPARDY!
$200
Answer:
It’s computed by finding the limit of the
difference quotient as ∆x approaches 0.
Question:
What is the derivative?
CALCULUS JEOPARDY!
$400
Answer:
It’s used to find the slope of a function
at a point.
Question:
What is the derivative?
CALCULUS JEOPARDY!
$600
Answer:
It’s used to find the slope of the tangent
line to a graph f (x), and is usually denoted
f’(x).
Question:
What is the derivative?
CALCULUS JEOPARDY!
$800
Answer:
It’s used to find the instantaneous rate of
change of a function.
Question:
What is the derivative?
The Derivative is…
computed by finding the limit of the difference
quotient as ∆x approaches 0.
the slope of a function at a point.
the slope of the tangent line to a graph f (x), and
is usually denoted f’(x).
the instantaneous rate of change of a function.
in conclusion...
The derivative is the the slope of the line
tangent to the curve (evaluated at a point)
it is a limit (3 ways to define it)
cool site to go to for additional
explanations:http://archives.math.utk.edu/visual.calculus/2/
