Unit #3 : Diﬀerentiability, Computing Derivatives...

Unit #3 : Differentiability, Computing Derivatives

Goals:

• Determine when a function is differentiable at a point

• Relate the derivative graph to the the graph of an original function

• Compute derivative functions of powers, exponentials, logarithms, and trig func-tions.

• Compute derivatives using the product rule, quotient rule, and chain rule forderivatives.

Textbook reading for Unit #3 : Study Sections 1.5, 2.3, 2.4, 2.6 and 3.1–3.6.

2

In the previous unit we introduced the definition of the derivative. In thisunit we will use and compute the derivative more efficiently. As a lead in, though,let us review how we arrived at the derivative concept.

Interpretations of Secants vs Derivatives

From Section 2.4The slope of a secant line gives

• the average rate of change of f(x) over some interval ∆x.

• the average velocity over an interval, if f(t) represents position.

• the average acceleration over an interval, if f(t) represents velocity.

Give the units of the slope of a secant line.

Unit 3 – Differentiability, Computing Derivatives 3

The derivative gives

• the limit of the average slope as the interval ∆x approaches zero.

• a formula for slopes for the tangent lines to f(x).

• the instantaneous rate of change of f(x).

• the velocity, if f(t) represents position.

• the acceleration, if f(t) represents velocity.

Give the units of the derivative.

4

Differentiability

From Section 2.6Recall the definition of the derivative.

f ′(x) =df

dx= lim

∆x→0

∆f

∆x= lim

h→0

f(x + h)− f(x)

h

A function f is differentiable at a given point a if it has a derivative at a, orthe limit above exists. There is also a graphical interpretation differentiability: ifthe graph has a unique and finite slope at a point. Since the slope inquestion is automatically the slope of the tangent line, we could also say that

f is differentiable at a if its graph has a (non-vertical)tangent at (a, f(a)).

For functions of the form y = f(x), we do not consider points with vertical tangentlines to have a real-valued derivative, because a vertical line does not have a finiteslope.


Here are the ways in which a function can fail to be differentiable at a point a:

1. The function is not continuous at a.

2. The function has a corner (or a cusp) at a.

3. The function has a vertical tangent at (a, f(a)).

Sketch an example graph of each possible case.

6

Investigate the limits, continuity and differentiability of f(x) = |x| at x = 0graphically.


Use the definition of the derivative to confirm your graphical analysis.

8

We have seen at a point that a function can have, or fail to have, the followingdescriptors:

• continuous;

• limit exists;

• is differentiable.

Put these properties in decreasing order of stringency, and sketch relevantillustrations.


Differentiability is Common

You will notice that, despite our concern about some functions not being differen-tiable, most of our standard functions (polynomials, rationals, exponentials, loga-rithms, roots) are differentiable at most points. Therefore we should investigatewhat all these possible derivative/slope values could tell us.

Interpreting the Derivative

From Section 2.4

• Where f ′(x) > 0, or the derivative is positive, f(x) is increasing.

• Where f ′(x) < 0, or the derivative is negative, f(x) is decreasing.

• Where f ′(x) = 0, or the tangent line to the graph is horizontal, f(x)has a critical point.

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AB

C

D EF G

Question 1 f ′(x) > 0 on the intervals

1. A− C, E −G

2. C − E

3. B −D, F −G

4. A− B, D − F


AB

C

D EF G

Question 2 f ′(x) takes on its largestnegative value at

1. A

2. B

3. C

4. D

5. G

12

Graphs, and Graphs of their Derivatives

From Section 2.3Example: Consider the same graph again, and the graph of its derivative.Identify important features that associate the two.


AB

C

D EF G

A B C D E F G

14

Question: Consider the graph of f(x) shown:

−1 0 1 2

−1

1

2

Which of the following graphs is the graph of the derivative of f(x)?


−1 0 1 2

−1

1

2

−1 0 1 2

−1

1

2

−1 0 1 2

−1

1

2

A B

−1 0 1 2

−1

1

2

−1 0 1 2

−1

1

2

C D

16

Note: The standard formulas for derivatives are covered in the Grade 12 Ontariocurriculum. While they will be reviewed here, students who are not familiar

with them should begin both textbook reading and the assignment

problems for this unit as soon as possible.

Computing Derivatives

From Sections 3.1-3.6Beyond the graphical interpretation of derivatives, there are all the algebraic rules.All of these rules are based on the definition of the derivative,

f ′(x) =df

dx= lim

∆x→0

∆f

∆x= lim

h→0

f(x + h)− f(x)

h

However, by finding common patterns in the derivatives of certain families of func-tions, we can compute derivatives much more quickly than by using the definition.


Sums, Powers, and Differences

Constant Functions:d

dxk = 0

Power rule:d

dxxp = pxp−1

Sums :d

dxf(x) + g(x) =

(

d

dxf(x)

)

+

(

d

dxg(x)

)

Differences:d

dxf(x)− g(x) =

(

d

dxf(x)

)

−(

d

dxg(x)

)

Constant Multiplier:d

dxkf(x) = k

(

d

dxf(x)

)

, k a constant

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Example: Evaluate the following derivatives:d

dx

(

x4 + 3x2)

d

dx

(

2.6√x− πx3 + 4

)


Question: The derivative of −3x2 − 1

x2is

1. −6x3 + 21

x3

2. −6x + 21

x3

3. −6x− 21

x3

4. −x3 + 21

x

20

Exponentials and Logs

e as a base:d

dxex = ex

Other bases:d

dxax = ax(ln(a))

Natural Log:d

dxln(x) =

1

x

Other Logs:d

dxloga(x) =

1

x

1

ln(a)



dx

(

4 · 10x + 10 · x4)

d

dx(ex + log10(x))

(Exponential and log derivatives are relatively straightforward, until we mix in theproduct, quotient, and chain rules.)

22

Product and Quotient Rules

Products:d

dxf(x) · g(x) = f ′(x)g(x) + f(x)g′(x)

Quotients:d

dx

f(x)

g(x)=

f ′(x)g(x)− f(x)g′(x)

(g(x))2


dx

(

4x2ex)


d

dx(x ln(x))

d

dx

(

5x2

ln(x)

)

24

Question: The derivative of10x

x3is

1.10x

ln(10)x−3 + 10x(−3x−4)

2.10x ln(10)x3 − 10x(3x2)

x6

3.10x 1

ln(10)x3 − 10x(3x2)

x6

4. ln(10)10xx−3 + 10x(−3x−4)


Chain Rule

Nested Functions:d

dxf(g(x)) = f ′(g(x)) · g′(x)

Liebnitz formd

dxf(g(x)) =

df

dg

dg

dx

26


dxex

2


d

dxln(x4)

28

d

dx

(

1

1 + x3

)


d

dx

(

x4 + 103x)

30

Question: The derivative of e√x is

1.1

2e

1√x

2. e√x(√

x)

3.1

2e√x

(

1√x

)

4.1

2e√x(√

x)


Trigonometric Functions

From Section 1.5In our earlier discussion of functions, we skipped over the trigonometric functions.We return to them now to discuss both their properties and their derivative rules.

The trigonometric functions are usually defined for students first using triangles(recall the mnemonic device, “SOHCAHTOA”).

32

Use the 45/45 and 60/30 triangles to compute the sine and cosine of thesecommon angles.


Extending Trigonometric Domains

One difficulty with limiting ourselves to the triangle ratio definition of the trigfunctions is that the possible angles are limited to the range θ ∈ [0, π2 ] radians orθ ∈ [0, 90] degrees.To remove this limitation, mathematicians extended the definition of the trigono-metric functions to a wider domain via the unit circle.

θ

34

How does the circle definition lead to the trigonometric identity sin2(θ) +cos2(θ) = 1?


Show how the circle and triangle definitions define the same values in the firstquadrant of the unit circle.

It is useful to understand both definitions of trig functions (circle and triangle) assometimes one is more helpful than the other for a particular task.

36

Sine and Cosine as Oscillating Functions

Despite the geometric source of the trigonometric functions, they are used morecommonly in biology and many other sciences as because their periodicity andoscillatory shapes. For many cyclic behaviours in nature, trigonometric func-tions are a natural first choice for modeling.


Question The graph of y = 10 + 4 cos(x) is shown in which of the followingdiagrams?

−6

−4

−2

2

4

6

8

10

12

14

2

4

6

8

10

12

14

A B

2

4

6

8

10

12

2

4

6

8

10

12

C D

Show the amplitude and the average on the correct graph.

38

Period and Phase

How can you find the period of the function cos(Ax)?


How can you reliably determine where the function cos(Ax + B) ‘starts’ onthe graph? (For a cosine graph, where the ‘start’ represents a maximum, thestarting time or x value is sometimes called the “phase” of the function.)

40

Consider the graph of the function y = 5 + 8 cos(π(x − 1)). What are thefollowing properties of the function:

• amplitude

• period

• average

• phase


Sketch the graph on the axes below. Include at least one full period of thefunction.

42

More complicated amplitudes

In the form y = A + B cos(Cx + D), the B factor sets the amplitude. In manyinteresting cases, however, that amplitude need not be constant.Sketch the graph of |y| = 5, and the graph of y = 5 cos(x) on the axes below.


Sketch the graph of |y| = x, and the graph of y = x cos(πx) on the axes below.Use only x ≥ 0

44

Use your intuition to sketch the graph of y = ex cos(πx) on the axes below.


Derivatives of Trigonometric Functions

From Section 3.5Having covered the graphs and properties of trigonometric functions, we can nowreview the derivative formulae for those same functions.The derivation of the formulas for the derivatives of sin and cos are an interestingstudy in both limits and trigonometric identities. For those who are interested,many such derivations can be found on the web1. However, it is in some ways moreuseful to derive the formula in a graphical manner.

1For example, http://www.math.com/tables/derivatives/more/trig.htm#sin

46

Below is a graph of sin(x). Use the graph to sketch the graph of its derivative.

−3 π /2 −π −π/2 0 π/2 π 3 π/2

−1

1

−3 π /2 −π −π/2 0 π/2 π 3 π/2

−1

1


From this sketch, we have evidence (though not a proof) that

Theoremd

dxsinx =

48

Most students will also be familiar with the other derivative rules for trig functions:

d

dxcos(x) = − sin(x)

d

dxtan(x) = sec2(x)

d

dxsec(x) = sec(x) tan(x)

d

dxcsc(x) = − csc(x) cot(x)

d

dxcot(x) = − csc2(x)


Prove the secant derivative rule, using the definition sec(x) =1

cos(x)and the

other derivative rules.

50

Question: Find the derivative of 4 + 6 cos(πx2 + 1)

1. 4− 6 sin(πx2 + 1) · (2πx)2. −6 cos(πx2 + 1) · (2πx)3. −6 sin(πx2 + 1) · (2πx)4. −6 sin(πx2 + 1) · (πx2 + 1)

5. 6 sin(2πx)


Inverse Trig Functions

From Section 1.5, 3.6In addition to the 6 trig functions just seen, there are 6 inverse functions as well,though the inverses of sine, cosine, and tangent are the most commonly used.Sketch the graph of sin(x) on the axes below

On the same axes, sketch the graph of arcsin(x), or sin−1 x, or the inverse ofsin(x).

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What is the domain of arcsin(x)?

What is the range of arcsin(x)?


In the next few questions you will obtain the formula for the derivative of arcsinx.

Simplify sin(arcsin x)

Differentiate both sides of this equation, using the chain rule on the left. You

should end up with an equation involvingd

dxarcsinx.

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Solve ford

dxarcsinx, and simplify the resulting expression by means of the

formula

cos θ =√

1− sin2 θ,

which is valid if θ ∈ [−π

2,π

2].

d

dxarcsinx =

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