Limit Definition of the
Derivative
Objective
 To
use the limit definition to find the
derivative of a function.
 TS: Devoloping a capacity for working
within ambiguity.
Slope
 Slope:
the rate at which a line rises or falls
 For
a line, the rate (or slope) is the same
at every point on the line.
 For
graphs other than lines, the rate at
which the graph rises or falls changes
from point to point.
Slope

This parabola is rising
more quickly at point
A than it is at point B.

At the vertex, point C,
the graph levels off.

At point D the graph
is falling.
Slope
 To
determine the rate at which a graph
rises or falls at a single point, we can find
the slope of the tangent line to the point.
 How
do we calculate the slope of a
tangent line?
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The Difference Quotient
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
f ( x  x)  f ( x)
f '( x)  lim
x0
x
provided this limit exists.
Derivatives

The derivative of the function y = f (x) may be
expressed as …
f '( x)
“f prime of x”
Prime notation
Leibniz notation
y'
“y prime”
dy
dx
“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:
f ( x)  x 2  3x  5
f ( x  x)  f ( x)
f '( x)  lim
x0
x
Limit Definition of the Derivative
f ( x)  x 2  3x  5
f ( x  x)  f ( x)
f '( x)  lim
x0
x
( x  x) 2  3( x  x)  5  ( x 2  3 x  5)
f '( x)  lim
x0
x
x 2  2 xx  (x) 2  3 x  3x  5  x 2  3 x  5
f '( x)  lim
x0
x
Limit Definition of the Derivative
2 xx  (x)  3x
f '( x)  lim
x 0
x
2
x(2 x  x  3)
f '( x)  lim
x0
x
f '( x)  2 x  3
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:
f ( x)  8 x 2  1
f ( x  x)  f ( x)
f '( x)  lim
x0
x
Limit Definition of the Derivative
f ( x)  8 x 2  1
f ( x  x)  f ( x)
f '( x)  lim
x0
x
8( x  x) 2  1  (8 x 2  1)
f '( x)  lim
x0
x
8( x 2  2 xx  (x) 2 )  1  8 x 2  1
f '( x)  lim
x0
x
Limit Definition of the Derivative
8 x 2  16 xx  8(x) 2  1  8 x 2  1
f '( x)  lim
x0
x
x(16 x  8x)
f '( x)  lim
x0
x
f '( x)  16 x
A formula for finding the
slope of the tangent line
of f (x) at a given point.
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?
CALCULUS JEOPARDY!
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Answer:
It’s the thing we love most about calculus.
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.