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Name: Defining the Derivative Section:

3.1 Defining the Derivative
Vocabulary Examples

Difference
Quotient For a function f , the difference quotient Q is:

Q =

Alternately, for h , 0, Q =

Slope of a
Tangent Line mtan =

Alternately, for h , 0, mtan =

Derivative of a
Function at a
Point

The derivative of f (x) at a, denoted , is defined:

f ′(a) =

Or f ′(a) =

Instantaneous
Rate of
Change

The instantaneous rate of change of a function f (x) at a is its

1. For each of the following functions, determine the slope of the secant line between x1 and x2.

(a) f (x) = 4x + 7, x1 = 2, x2 = 5

(b) f (x) = xx+3 , x1 = 0, x2 = 3

2. For each of the following functions, determine the f ′(a)

(a) f (x) = 2×2 − x, a = 4

(b) f (x) =

x − 7, a = 10

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Name: Defining the Derivative Section:

3. For each of the following functions f , write the equation for the line tangent to f at x = a

(a) f (x) = 13 x
5 + 2x at a = 1

(b) f (x) = 4√
x

at a = 2

(c) f (x) = 54
x3

+ 5 at a = −3

4. Recall that the velocity of a moving object is instantaneous rate of change of its position. A projectile’s
position d at time t is given by the function d(t) = −4.9t2 + 20.1x + 24.3.

(a) Determine the velocity of the object after 2 seconds.

(b) Determine the velocity of the object after 3 seconds.

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Name: Derivative as a Function Section:

3.2 Derivative as a Function
Vocabulary Examples

Derivative
Function For a function f , the derivative function, denoted ,

is the function whose domain consists of values of x such

that the following limit exists:

f ′(x) =

Notations:

Theorem on
Differentiabil-
ity and
Continuity

If a function f is differentiable at a, then f is at a.

Higher-Order
Derivative

The of a

1. Use the definition of a derivative to determine the derivative of the following functions.

(a) f (x) = 3×2 − 2

(b) f (x) = x−2

(c) f (x) =

3x − 7

(d) f (x) = 3√
x

2. Use the graph of each of the following functions to sketch the graph of its derivative.

(a)

−2 2

−4

−2

2

4

(b)

−4 −2 2 4

−4

−2

2

4

(c)

−2 2

−4

−2

2

4

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Name: Derivative as a Function Section:

3. The second derivative f ′′(x) = lim
h→0

f ′(x+h)− f ′(x)
h . Determine f

′′(x) for each of the following
functions.

(a) f (x) = 17 x + 7

(b) f (x) = −4.9×2 − 7x + 121.9

(c) f (x) = (x + 1)3

4. Velocity is the first derivative of the position (or displacement) function. Acceleration is the second
derivative of the position (or displacement) function. Consider a particle whose position can be de-
scribed by the function d(t) = 11.2t2 + 3t − 10. Using the definition of the derivative, determine
(a) the function that models the velocity of the particle and (b) the acceleration of the particle.

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Name: Differentiation Rules Section:

3.3 Differentiation Rules
Vocabulary Examples

Constant Rule
For any constant c, dd x (c) =

Power Rule
d
d x (x

n) =

Constant
Multiple Rule For any constant c and differentiable function f ,

d
d x (c · f (x)) = c·

Sum Rule
d
d x ( f (x) + g(x)) =

Difference
Rule d

d x ( f (x) − g(x)) =

Product Rule
d
d x ( f (x) · g(x)) =

Quotient Rule
d
d x

(
f (x)
g(x)

)
=

1. Determime f ′(x) for each of the following functions.

(a) f (x) = 13 x
6 − 3×1/3 + 10

x3
(b) f (x) = (x + 2)(2×2 − 3) (c) f (x) = 4x

3−2x+1
x2

2. The following graph shows f (x) and g(x). h(x) = f (x) + g(x).

2 4

2

4 f (x)

g(x)

(a) Determine h′(1)
(b) Determine h′(3)
(c) Determine h′(4)

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Name: Differentiation Rules Section:

3. Assume, for each of the following, that f and g are both differentiable. Determine h′(x).

(a) h(x) = 4 f (x) + g(x)7 (b) h(x) = x
3 f (x) (c) h(x) = f (x)g(x)2

4. Find the equation of the line tangent to the graph
of f (x) = x2 + 4x − 10 at x = 8

5. Find the equation of the line tangent to the graph

of f (x) = 2x
7/3−3×6+x

x2
at x = −1

6. Find the equation of the line tangent to the graph
of f (x) = (3x − x2)(3 − x − x2) at x = 1

7. Find the equation of the line tangent to the graph of
f (x) = 6x−1 at and containing the point (1,−6)

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Name: Differentiation Rules Section:

8. A car driving along a freeway with traffic has
traveled d(t) = t3 − 6t2 + 9t meters in t seconds.

(a) Determine the time, in seconds, when the
velocity of the car is 0.

(b) Determine the acceleration of the car when
the velocity is 0.

9. The concentration of antibiotic in the bloodstream
t hours after being injected is given by the function
C(t) = 2t

2+t
t3+50

, where C is measured in miligrams
per litre of blood.

(a) Find the rate of change of C(t).
(b) Determine the rate of change for t = 8,

t = 12, t = 24.
(c) Describe what is happening as the number of

hours increases.

10. Determine a quadratic function for which f (1) = 5, f ′(1) = 3, and f ′′(1) = −6

For use with OpenStax Calculus, free at https://openstax.org/details/books/calculus-volume-1 31

3.1 Defining the Derivative
Vocabulary Examples

Difference Quotient

For a function f , the difference quotient Q is:

Q =
Alternately, for h * 0, Q =

Slope of a
Tangent Line

mtan =
Alternately, for h * 0, mtan =

Derivative of a
Function at a Point

The derivative of f (x) at a, denoted , is defined:

f ,(a) =
Or f ,(a) =

Instantaneous
Rate of Change

The instantaneous rate of change of a function f (x) at a is its

1. For each of the following functions, determine the slope of the secant line between x1 and x2. (a) f (x) = 4x + 7, x1 = 2, x2 = 5

Name:
Defining the Derivative
Section:

For use with OpenStax Calculus, free at https://openstax.org/details/books/calculus-volume-1
25

(b) f (x) =

x x+3

, x1 = 0, x2 = 3

2. For each of the following functions, determine the f ,(a)
(a) f (x) = 2×2 − x, a = 4

(b) f (x) = √x − 7, a = 10

3. For each of the following functions f , write the equation for the line tangent to f at x = a

(a) f (x) = 1 x5 + 2x at a = 13

(b) f (x) =
x

√4 at a = 2

(c) f (x) = 54 + 5 at a = −3
x3

4. Recall that the velocity of a moving object is instantaneous rate of change of its position. A projectile’s position d at time t is given by the function d(t) = −4.9t2 + 20.1x + 24.3.
(a) Determine the velocity of the object after 2 seconds.
(b) Determine the velocity of the object after 3 seconds.

3.2 Derivative as a Function
Vocabulary Examples

Derivative
Function

For a function f , the derivative function, denoted , is the function whose domain consists of values of x such that the following limit exists:

f ,(x) =

Notations:

Theorem on
Differentiabil- ity and Continuity

If a function f is differentiable at a, then f is at a.

Higher-Order
Derivative

The of a

1. Use the definition of a derivative to determine the derivative of the following functions. (a) f (x) = 3×2 − 2
(b) f (x) = x−2
(c) f (x) = √3x − 7

(d) f (x) = 3√

x

2. Use the graph of each of the following functions to sketch the graph of its derivative.

Name:
Derivative as a Function
Section:

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27

(a)

(b)

(c)

4
2
−2
−2
−4
2
4
2
−4
−2
2
4
−2
−4
4
2
−2
−2
−4
2

3. The second derivative f ,,(x) = lim

h→0
h

f ,(x+h)− f ,(x). Determine f ,,(x) for each of the following

functions.
(a) f (x) = 1 x + 77

(b) f (x) = −4.9×2 − 7x + 121.9
(c) f (x) = (x + 1)3

4. Velocity is the first derivative of the position (or displacement) function. Acceleration is the second derivative of the position (or displacement) function. Consider a particle whose position can be de- scribed by the function d(t) = 11.2t2 + 3t 10. Using the definition of the derivative, determine−

(a) the function that models the velocity of the particle and (b) the acceleration of the particle.

3.3 Differentiation Rules
Vocabulary Examples

Constant Rule

For any constant c, d

(c) =

dx

Power Rule

d

(xn) =

dx

Constant
Multiple Rule

For any constant c and differentiable function f ,

d

(c · f (x)) = c·

dx

Sum Rule

d

( f (x) + g(x)) =

dx

Difference Rule

d

( f (x) − g(x)) =

dx

Product Rule

d

( f (x) · g(x)) =

dx

Quotient Rule

d f (x) =

dx g(x)

1. Determime f ,(x) for each of the following functions.
x2

Name:
Differentiation Rules
Section:

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29

(a) f (x) = 1 x6 − 3×1/3 + 10
3

x3

(b) f (x) = (x + 2)(2×2 − 3) (c) f (x) = 4×3−2x+1

2. The following graph shows f (x) and g(x). h(x) = f (x) + g(x).
(a) Determine h,(1)(b) Determine h,(3)
4

f (

x)

2

g(x

)

2

4

(c) Determine h,(4)

3. Assume, for each of the following, that f and g are both differentiable. Determine h,(x).2

(a) h(x) = 4 f (x) +
g

(x)
7

(b) h(x) = x3 f (x) (c) h(x) = f (x)g(x)

4. Find the equation of the line tangent to the graph of f (x) = x2 + 4 − 10 at x = 8
x

5.
Find the equation of the line tangent to the graph of f (x) = 2×7/3−3×6+x at x = −1

x2

6. Find the equation of the line tangent to the graph 7. Find the equation of the line tangent to the graph of
of f (x) = (3x − x2)(3 − x − x2) at x = 1 f (x) = x61 at and containing the point (1, −6)−

8. A car driving along a freeway with traffic has 9. The concentration of antibiotic in the bloodstream

traveled d(t) = t3 − 6t2 + 9t meters in t seconds.
(a) Determine the time, in seconds, when the velocity of the car is 0.
(b) Determine the acceleration of the car when the velocity is 0.

t hours after being injected is given by the function

C(t) = 2t2+t , where C is measured in miligrams
t3 50

+
per litre of blood.
(a) Find the rate of change of C(t).
(b) Determine the rate of change for t = 8,

t = 12, t = 24.
(c) Describe what is happening as the number of hours increases.

10. Determine a quadratic function for which f (1) = 5, f ,(1) = 3, and f ,,(1) = −6

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