AP Calculus AB/BC
Unit 2: Introduction to the Derivative and the Tangent Line
Target 2.1 The student will be able to demonstrate an understanding of the difference between average and
instantaneous rates of change, describe the concept of a numerical derivative, and use a variety of
notations for derivatives.
Target 2.2 The student will be able to use limits to find derivatives analytically.
Target 2.3 The student will be able to approximate derivatives numerically and graphically, and evaluate
derivatives using technology.
Target 2.4 The student will be able to find the tangent and normal lines to a curve at a specific point.
Target 2.5 The student will be able to describe the derivative of a function given the graph of that function,
and will be able to describe a function given the graph of its derivative.
Derivative Notation (from p.101 in text)
f ' ( x)
y'
dy
dx
df
dx
d
f (x )
dx
“ f prime of x ”
“ y prime”
“ dy dx ” or “the derivative of y with respect to x ”
“ df dx ” or “the derivative of f with respect to x ”
“the derivative of f at x ”
Average Rate of Change = (amount of change)  (length of interval) = slope!
y 2  y1
x 2  x1
Secant Line – a line that goes through two points on a graph
Tangent Line – a line that contains one point on a graph, and goes along the graph at that point, but does not
cross through the graph at that point. (It may cross the graph at a different point depending on the shape of
the graph.)
Our goal is to turn the secant line into a tangent line by moving one point on the secant line infinitely close to
the other.
AP Calculus AB/BC
Unit 2: Introduction to the Derivative and the Tangent Line
Slope of the Secant Line between the points P  (a, f (a)) and Q  (a  h, f (a  h)) is
f ( a  h)  f ( a ) f ( a  h)  f ( a )
m

( a  h)  a
h
By moving one of the secant points closer and closer to the other secant point, we make the distance between
those two points (which is “h”) get smaller and smaller. The idea is that we get “h” to be as close to 0 as
possible. So we turn this slope formula into a limit:
Slope of the Tangent Line at x  a is m  lim
h0
f ( a  h)  f ( a)
h
By using the slope of the tangent line, we can find the instantaneous slope of the function.
The slope of the tangent line to the graph of f ( x ) at x  a is also called the derivative of f at a . If a
function has a derivative, then the function is differentiable. For a function to be differentiable, its left- and
right-handed derivatives must be equal at each x -value.
f ( x  h)  f ( x )
(This form gives a generic slope of a graph at
h 0
h
any value of x and may also be used to evaluate slope at x  a .)
The definition of the derivative is f ( x)  lim
Alternate Definition of the Derivative of f at x  a :
f ' ( a )  lim
xa
f ( x)  f (a)
xa
AP Calculus AB/BC
Equation of the Tangent Line
Unit 2: Introduction to the Derivative and the Tangent Line
There are many forms of linear equations. However, we will use point-slope form most often:
y  y1  m( x  x1 )
In function and derivative notation, this looks like:
y  f ( a )  f (a )( x  a )
Note: On multiple choice exams, tangent lines may be given in any linear form, including slope-intercept form
y  mx  b and standard form Ax  By  C .
A normal line is the line perpendicular to the tangent
line at the point of tangency.
You can evaluate a derivative using your graphing calculator! Enter the function in your y  screen. Press
GRAPH. Then go to 2nd CALC (above TRACE), and select 6: dy dx . This returns you to the graph. Enter the
value of x , and your screen will display the value of the derivative. This is a great way to check your answer
when using algebraic methods.
AP Calculus AB/BC
Unit 2: Introduction to the Derivative and the Tangent Line
Target 2.6 The student will be able to define differentiability at a point, understand the relationship
between continuity and differentiability in a function, and describe common types of non-differentiability in
the graph of a function.
A derivative (instantaneous slope) does not exist at a point for any of the following reasons:
Corner
Cusp
Example: y  x
Vertical tangent
Example: y  x
2
Example: y  x 3
1
3
Discontinuity
Example: y  [| x |]
Differentiability implies continuity. (If a graph has a derivative at a point, then it is continuous at that point.)
However!!! Continuity does not imply differentiability. (Just because a graph is continuous at a point, it
doesn’t mean it has a derivative at that point.)
AP Calculus AB/BC
Unit Two References
Unit 2: Introduction to the Derivative and the Tangent Line
Essential Skill
2.1 The student will be able to demonstrate an
understanding of the difference between average and
instantaneous rates of change, describe the concept of a
numerical derivative, and use a variety of notations for
derivatives.
2.2.a The student will be able to find derivatives
f ( a  h)  f ( a )
analytically using the limit definition lim
h0
h
2.2.b The student will be able to find derivatives
analytically using the alternate definition
f ( x)  f (a )
lim
xa
xa
2.3 The student will be able to approximate derivatives
numerically and graphically, and evaluate derivatives
using technology.
2.4 The student will be able to find the tangent and
normal lines to a curve at a specific point.
2.5 The student will be able to describe the derivative of
a function given the graph of that function, and describe
a function given the graph of its derivative.
2.6 The student will be able to define differentiability at a
point, understand the relationship between continuity
and differentiability in a function, and describe common
types of non-differentiability in the graph of a function.
Text Examples
p.92 #1, 7-8, 13-14
p.105 #21, 23
p.105 #1-4, 9-12
p.92 #15-17, 19-22, 29-30
p.105 #5-8
p.92 #9-12
p.105 #17-20
p.105 #13-16, 22, 26-28
p.114 #1-2, 5-10, 12-26, 31-36, 39