• To differentiate transcendental functions and apply these derivatives to graphing functions and solving word problems containing transcendental functions. Supported by learning objectives 1 through 4.
| G.1 |
| G.2 |
| G.3 |
| G.4 |
Objective
G.1 Back to Top
To differentiate logarithmic functions and solve related problems.
Instructional Notes
Differentiation to this point in the course has been limited to algebraic functions (see B.2). In this unit, differentiation skills will be extended to the transcendental functions. This objective deals with logarithmic functions. It is through this objective that students will be introduced to e . They may have had a small taste of e in C.2, depending on whether or not you chose to do the related exercise.
Since logarithms were last studied in Math B30, it will be necessary to review the definition of a logarithm, the properties of logarithms (including the change of base formula), the graphs of logarithmic functions, and the domain and range of logarithmic functions.
Reintroduce students to logarithms by recalling that 
because 
. This simple statement is extremely powerful in helping students remember how to convert back and forth from logarithmic to exponential statements as well as to recall the definition of a logarithm.
Definition of a logarithm: More formally, 
if and only if 
. b is the base of the logarithm (must be positive, but not 1), N is the number whose logarithm we are finding (it must be positive), and k is the logarithmic value or exponent to which b must be raised in order to obtain N .
1. 
2. 
3. 
4. 
5. 
6. 
7. 
(The change of base formula)
It would be wise to practice these properties orally using specific values. For example: 
, 
, 
, 
, 
, 
.
x |
|
|
10 |
omit |
1 |
8 |
3 |
.90 |
4 |
2 |
.60 |
2 |
1 |
.30 |
1 |
0 |
0 |
|
|
|
|
|
omit |
|
|
omit |
|
omit |
|
x |
|
8 |
|
4 |
|
2 |
|
1 |
0 |
|
1 |
|
2 |
|
3 |
Students should sketch the graphs of 
, 
, and 
by creating a suitable table of values.
From the definition of a logarithm, and from the graphs shown, it can be seen that the domain of 
is 
while the range is 
. Note, too, that if the base, b , lies between 0 and 1, 
is a decreasing function, otherwise it is an increasing function.
In the process of deriving the formula for the derivative of 
, students will encounter 
. If they have not done activity 1 in C.2, have them complete a table similar to the one below with the aid of a calculator.
What is happening to the value of 
as k approaches 0? Students should observe that it is approaching a limiting value of 2.71828…. Point out that the number being approached is used so frequently in calculus that it is designated by the letter e , much as 
is used to represent the ratio of the circumference of a circle to its diameter. Thus, 
. This number is irrational although it looks rational because the digits 1828 do repeat initially.
To develop the formula for the derivative of 
, return to the definition of the derivative:
the definition of the derivative:
applying the definition to the function:
using log property 2:
reorganizing the expression:
If we let 
, then 
, and 
. Note that if 
, so will 
since 
, and x cannot be 0 since 0 is not in the domain of the function. We can rewrite the limit as shown. Note also the use of log property 3.
Based on the table work, 
. This gives 
.
At this point, it would be wise to do an example that uses this result.
Example: Find the slope of the tangent line to the graph of 
at the point where 
. Give the answer to four decimal places.
Solution: If 
, then 
. Thus, 
. To evaluate 
, the change of base formula will be needed since the calculators are not programmed to give logarithms in base 2 but rather in base 10. Thus, 
.
Next, generalize the result so that you can find the derivative of 
where u is a function of x . Using the chain rule, we have:
If 
, 
Putting this result into words, every time you use it, will greatly enhance students' ability to remember it. “To find the derivative of the logarithm of an expression in base b , take the reciprocal of the expression, multiply that by the logarithm of e in base b , and multiply that by the derivative of the expression.”
Example: If 
, find 
.
Solution: 
.
Point out to the students that to actually use this result to find the slope of the tangent line at a particular x value, you have to use the change of base formula to evaluate 
.
Next, point out to the students that using the properties of logarithms may make differentiation of logarithmic functions easier. Consider the next example done two different ways – one by simplifying first with log properties, the other without.
Example: If 
, find 
.
Solution:
| Using log properties | Not using log properties |
 |
 |
 |
 |
Example: If 
, find 
.
Solution: Since 
is a constant (note there is no variable), 
.
Having derived a formula for the derivative of logarithmic functions in base b , introduce students to logarithmic functions in base e or natural logarithmic functions. Mention that just as 
is written as 
, so too 
is written as 
for convenience.
We know that if 
, then 
. Thus, if 
, (note that 
) then 
. Using e as the base of our logarithms eliminates the tag-a-long 
. Thus we have established that if 
, then 
.
Introduce the graph of 
. Superimpose it over the graphs of 
, 
, and 
made earlier. Note that the domain and range of 
is the same as the domain and range for any function of the form 
, as it, too, is of that form.
Now, have the students find the slope of the tangent line to 
at each of the x values used to create its graph. Since 
, we have the results shown in the table at left. See how easy it is to find the slope of the tangent line to the natural logarithmic function compared to logarithmic functions in other bases.
After students are comfortable finding the slope of the tangent line to 
, introduce them to finding the slope of the tangent line to 
where u is a function of x . Make sure the students recognize that the chain rule is being applied in finding 
.
If 
, then 
.
Finally, with student input, complete some examples similar to the following.
Example: If 
, find 
.
Solution: 
.
Example: If 
, find 
.
Solution: 
Example: If 
, find 
.
Solution: 
.
Example : If 
, find 
.
Solution: Use log properties to rewrite 
. Thus, 
. Then differentiate: 
.
Occasionally, students are unsure when to use the product rule if logarithmic functions are a part of the question. A pair of examples, like the following, should help to clarify the issue.
Example: If 
, find 
.
Solution: 
. Note that no product rule was involved. It could have been used, but 10's derivative would have been 0. Remind students that to differentiate 
, no product rule was used. A product rule should be used if the both terms in the product contain a variable .
Example: If 
, find 
.
Solution: A product rule is required since both terms in the product contain a variable. 
.
When students know how to differentiate natural logarithmic functions, students should be required to apply these skills to curve sketching and problem solving. Consider the following examples.
Example: Find the relative extrema of the function 
.
Solution: Note that the domain is 
– we can only find logarithms of positive numbers. 
. The potential critical numbers are 
, 
(they make 
zero), and 
(which makes 
undefined). Both 
and 
are not in the domain of the function. Thus only 
is a critical number.
Performing a sign analysis on 
yields:
Since around 
is 
, 
or 
is a local minimum point.
Example: After the discovery of a rich natural gas deposit, a community's population increased quickly and could be approximated by the function 
, where x is the number of years after the discovery.
(a) What was the population at the time of the discovery?
(b) What was the population 2 years after the discovery?
(c) Find 
and interpret the result.
Solution:
(a) At the time of the discovery, 
. Thus the population was 
people.
(b) After 2 years, the population was 
people.
(c) 
. Thus, 
. 2 years after the discovery, the population was growing at a rate of 1600 people/year.
Examples/Activities
Students should practise working with the log properties. Plan exercises similar to the following.
1. Evaluate each of the following logarithms, if possible.
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
(g) 
(h) 
2. Use a calculator, together with the change of base formula where necessary, to find each of the following logarithms rounded to 5 decimal places.
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
(g) 
3. Write each of the following as the logarithm of a single simplified expression. The answer must be in the form 
, where o is the simplified expression.
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
4. Expand each of the following logarithmic expressions in terms of 
, 
, and 
.
(a) 
(b) 
(c) 
(d) 
When students have been introduced to natural logarithms, have students perform operations similar to the ones below.
1. Evaluate each of the following logarithms, if possible.
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
(g) 
(h) 
2. Use a calculator to find the value of each of the following rounded to 5 decimal places.
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
3. Write each of the following as the natural logarithm of a single simplified expression. The answer must be in the form 
, where o is the simplified expression.
(a) 
(b) 
(c) 
(d) 
(e) 
4. Expand each of the following logarithmic expressions in terms of 
, 
, and 
.
(a) 
(b) 
(c) 
(d) 
The majority of differentiation that students perform in this section will be with natural logarithmic functions. It is important, however, that students be able to differentiate logarithmic functions whose bases are other than e .
1. Find the derivative of each of the following functions.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h).
2. Find the slope of the tangent line to the given function at the given value of x . Round the answer to five decimal places.
(a)
at
(b)
at
.
3. Find the equation of the tangent line to the graph of 
drawn at the point where 
. Leave 
in unsimplified form in your answer.
Differentiating natural logarithm functions occurs much more frequently than other logarithmic bases.
Find the derivative of each of the following functions.
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. Use implicit differentiation to find 
if 
.
21. Consider the function 
.
(a) What is the domain and range of this function?
(b) Find 
.
1. Suppose that the number of units of a commodity that are sold after spending x hundred dollars on advertising is given by the function 
.
(a) How many units are sold if $100 is spent on advertising? Round to the nearest integer.
(b) How many units are sold if $1000 is spent on advertising? Round to the nearest integer.
(c) Find 
.
(d) Find 
and interpret the result.
(e) Find 
and interpret the result.
(f) Explain why 
?
(g) Does 
have any local extrema? Explain.
2. A man's bank account after x months shows a balance given by the function 
.
(a) How much money was in his account initially?
(b) How much money was in his account after 10 months?
(c) Find 
.
(d) Find 
and interpret the result.
(e) How many months did it take for the account to go down to $0?
3. A salesperson at a car dealership began work in an unfamiliar city. The number of people whose names the salesperson could remember after working x weeks is given by the function 
.
(a) How many people did the salesperson know initially?
(b) How many people did the salesperson know after 4 weeks? Round to the nearest integer.
(c) Find 
.
(d) Find 
and 
. Interpret your results.
(e) As x increases, does 
increase or decrease? What might be the reason?
For each of the following functions, determine the equation of the tangent line drawn at the point 
.
1. 
2. 
3. 
4. 
5. 
For each of the following functions, determine the equation of the tangent line drawn at the point 
. Leave e in the answer.
1. 
2. 
3. 
For each of the following functions, determine the critical numbers and the local extrema.
1. 
2. 
3. 
4. 
5. 
6. 
Find the intervals in which the function is concave up and concave down.
1. 
2. 
Suggestions/Extensions
Have students discuss questions such as: Why are logarithms necessary? Why can't logarithms have a base of 1?
Have the students research real-life occurrences of e and ln x to bring further meaning to students study.
Have the students discuss how similar looking expressions are actually different. For example, how are 
and 
different?
Provide an opportunity for the students to share different strategies for differentiating logarithmic expressions. Ask the students to identify the log rules they are using in their solutions.
When the students have understood 
when 
, have them revisit some of the different logarithm base questions that were done previously and use knowledge of ln x to find the derivatives another way.
Logarithmic Differentiation
Earlier in the course, we have seen that if 
, where n is a rational number, then 
. We cannot find the derivative of a function such as 
in the same way since the exponent is also a variable and not a fixed rational number. We can, however, find the derivative of such a function by taking the natural logarithm of both sides, before differentiating implicitly relative to x . This is known as logarithmic differentiation.
If 
, then 
.
Thus 
.
Differentiating implicitly with respect to x , we have:
Since 
, we have 
.
Use logarithmic differentiation to differentiate each of the following functions.
1. 
2. 
3. 
The above are all examples of how mathematicians change the representation in situations to create something to work with more readily. Have the students reflect upon their experiences with this strategy used in mathematics.
Logarithmic differentiation is also useful even if the variable does not appear in the exponent. Consider the following example.
Encourage students to try logarithmic differentiation on prior questions that have done.
The following theorem is useful for integration purposes later in the course.
If 
, then 
.
Proof:
If 
, then 
, and the result follows as was proven earlier in this objective.
If 
, then 
. Thus, 
. Then, 
Objective
G.2 Back to Top
To differentiate exponential functions and solve related problems.
To engage in activities that require exploration in order to develop understandings of a concept (CCT).
To further develop as independent learners within a classroom environment that promotes self-esteem, curiosity, competence, and trust (IL).
To use language as a tool for learning and communicating (COM).
Instructional Notes
Have students recall the definition of an exponential function and its graph. Students have studied these functions in Math B30. It is critical to point out the difference between a power function and an exponential function. Where is the variable located? That is the key.
An exponential function is a function that can be written in the form 
where b , the base, is a positive number other than 1.
Point out that since the base, b , can be any positive number, it could also be the value e , obtained in the limit process of finding the derivative of logarithmic functions. Thus 
, 
, 
, and 
are all examples of exponential functions. Using a table of values, or a graphing calculator, have the students draw the graphs of these four exponential functions. Have students study the graphs to observe that for bases greater than 1, the functions are increasing. Ask students how the size of the base determines the position of each graph relative to the others.
In Math B30, students have been introduced to inverse functions including the fact that logarithmic and exponential functions are inverses. For example, if 
, then the inverse of this function, obtained by interchanging x with y , can be written as 
. Solving for y by using the definition of a
logarithm, we have 
, an exponential function with base 2.
Thus, 
and 
are inverses of one another. Point out how the graphs of these functions are mirror images of one another about the line 
.
Discuss the domain and range of exponential functions. 
has domain 
and range 
.
The derivative of the exponential function is easily obtained without having to use limits. The students should be able to arrive at the result with only a little prompting ( CCT , IL ).
Have students begin with 
. Point out that we cannot find 
by writing 
since 
is an exponential function (variable in the exponent) not a power function (variable in the base).
Suggest to the students that they take the natural logarithm of both sides of 
and then differentiate implicitly to find 
. Some students will be able to do so on their own. Their work should progress as follows:
Thus if 
, 
.
Immediately following the development of this result, ask the students to find the slope of the tangent line to 
at 
. They should obtain 
and thus 
. Have students refer back to the graph of 
earlier and see if this is a reasonable value.
Next, ask the students to generalize the above result to give the derivative of 
where u is a function of x .
If 
, then 
Students should now try this generalized result with examples such as the following.
Example: Find 
if 
.
Solution: 
.
Example: Find 
if 
.
Solution: Using the product rule, 
Point out to the students that differentiating 
with respect to x results in a factor of 
. Ask students if they can think of an exponential function with a base that would cause that factor to be eliminated. Hopefully, they will respond with 
. Have them find the derivative of 
.
If 
, then 
. Since 
, 
.
Thus if 
, 
Note that the derivative of the function is the same as the function.
Have the students determine the slope of the tangent line to the graph of 
at 
. 
and thus 
. Ask the students to compare this result to the slope of the tangent line to 
at 
as determined earlier. The larger slope, 
, is consistent with the relative shapes of the graphs of 
and 
at 
.
Next, ask the students to generalize the result obtained above to find the derivative of 
where u is a function of x .
If 
, 
Have students try out this result with a few examples similar to the following.
Example: If 
, find 
.
Solution: 
.
Example: Find 
if 
.
Solution:
The following properties of logarithmic and exponential functions should be established. Properties are most useful in simplifying functions thus making differentiation easier. You could give students the left side of the statement and see if they could determine what the right side should be through experimentation with specific x values.
Give the students an opportunity to practise these properties.
Example: Simplify each of the following:
(a) 
(b) 
(c) 
(d) 
(e) 
Solution:
(a) 
using property 1.
(b) 
using property 2.
(c) 
using property 3.
(d) 
using property 4.
(e) 
. Note that before property 4 can be applied, the 
must be moved using the logarithmic property 
.
Next, have the students differentiate functions that can first be simplified using these four properties. See the following examples.
Example: Find 
if 
.
Solution: 
.
Example: Find 
if 
.
Solution: 
.
Have the students solve applied problems and graphing problems related to exponential functions, discussing their solutions in pairs (COM).
Examples/Activities
Use your calculator to give the value of each of the following rounded to 5 decimal places.
1. 
2. 
3. 
4. 
5. 
6. 
Starting with the graph of 
, sketch the graph of each of the following functions using a shift, reflection, stretch, or compression.
1. 
2. 
3. 
4. 
5. 
6. 
7. 
What is the domain and range of each of the following functions? Determine these results without the use of a graphing calculator. Confirm with a graphing calculator.
1. 
2. 
3. 
Differentiate each of the following functions.
1. 
2. 
3. 
4. 
5. 
6. 
(Use the product rule)
7. 
8. 
Find the slope of the tangent line to the given curve at the given point. Express your answer to 5 decimal places.
1. 
at 
2. 
at 
3. 
at 
4. 
at 
Find the derivative of each of the following functions.
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 
17. 
18 Find an expression for the 100 th derivative of 
.
19. Find the equation of the tangent line drawn to the given function at the given point. Leave e in your answer where applicable.
(a) 
, 
(b) 
, 
(c) 
, 
Simplify each of the following without the use of a calculator.
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
Find the derivative of each of the following functions.
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. For the functions 
and 
, find:
(a) the interval(s) in which 
is increasing and decreasing.
(b) any local extrema.
(c) the interval(s) in which 
is concave up and concave down.
11. For the function 
, find:
(a) the intervals in which 
is increasing and decreasing.
(b) the critical numbers.
(c) any local extrema.
12. Open garbage attracts rodents. Suppose that the number of mice in a neighbourhood, t weeks after a strike by garbage collectors, can be approximated by the function 
.
(a) How many mice are initially in the neighbourhood?
(b) How long does it take for the population of mice to quadruple?
(c) How many mice are in the neighbourhood after 5 weeks?
(d) How long does it take until there are 1000 mice?
(e) Find 
and interpret the result.
13. Suppose that the temperature of a cup of coffee in degrees Celsius, t minutes after it is poured, is given by the function 
.
(a) What was the initial temperature of the coffee?
(b) When was the coffee 
C?
(c) What was the temperature after 11 minutes?
(d) Find 
.
(e) Find 
and interpret the result.
(f) Find 
and interpret the result.
14. After t years, a motorcycle manufacturer has sold a total of y thousand units where 
. If 1500 units were sold after 1 year, find:
(a) the value of k . (Hint: 
when 
, solve for k .)
(b) how many units were sold after 5 years?
(c) 
.
(d) 
and interpret the result.
Suggestions/Extensions
Have the students research natural occurrences of e (IL).
When students are showing or discussing their solutions, encourage sharing of strategies and identification of rules or concepts being applied.
Neglecting inhibiting or stimulating factors, a population will normally grow at a rate proportional to its size. Thus, if 
is the population after time t , then the relationship between the growth rate and the current population size can be expressed by the equation 
where k is a constant.
Exponential functions with base e behave this way. If, for example, the population of a city is given by the function 
, then 
. Note that if 
, 
. Thus, 
is the initial population, or population at time 0. The constant k is the growth rate factor.
Example: If a city's growth rate is 6% per year, and the city's population is currently 75000, find:
(a) the equation of population growth.
(b) the population in 10 years.
Solution:
(a) We are told that 
. Since the population can be modelled by the function 
and since 
and 
, we have 
.
(b)
Have the students research for other growth rate functions and discuss what the derivative of those functions would represent (IL, COM).
Objective
G.3 Back to Top
To evaluate limits involving trigonometric functions and to apply the Squeeze Theorem appropriately.
Instructional Notes
In order to find the derivatives of the trigonometric functions in section G.4, two limits need to be evaluated, namely 
and 
. To evaluate these limits, requires an understanding of the Squeeze Theorem.
Ask the students how they, with a vehicle, might try to bring to a halt a trespasser who is driving on their land without threatening him/her with a gun, shooting out his/her tires, or ramming the vehicle. Would it be easier to bring the trespasser to a halt using two vehicles? If so, how?
It might be difficult for one driver to bring the trespasser to a halt, but if there were two drivers, they could come alongside the trespasser and sandwich him/her between their vehicles forcing him/her to stop – especially if the trespasser did not want his/her vehicle dented.
The Squeeze Theorem concerns the limit of a function that is squeezed or sandwiched between two other functions, both of which have the same limit at a given value for x as shown in the figure below. Proof of the Squeeze Theorem can be omitted.
The Squeeze TheoremIf when x is near c (except perhaps when x is at c itself), 
, and if 
, then 
.
You will need The Squeeze Theorem to determine 
.
It is critical to point out to students that the proof that follows involves measurement of 
in radians only . All the derivative formulas that are developed for trigonometric functions in G.4 are valid only if the limits involve angles that are measured in radians.
Consider the unit circle at right with angle 
drawn in standard position.
By examining the figure, we see that the area of 
is less than the area of sector POB which in turn is less than the area of 
.
Area 
: Since 
is measured in radians, the coordinates of P are 
– well known from Math C30. Thus, the area of 
.
Area Sector POB : The area of a sector with central angle 
radians and radius r is 
. Since this is a unit circle, 
. Thus sector POB has area 
.
Area 
: 
has area 
. OB is the radius of the unit circle and is thus 1. 
is a right triangle and thus 
. Thus 
. Therefore 
has area 
.
Now putting it all together, we have:
Notice that 
is squeezed between 
and 
, both of which approach 1 as 
(direct substitution). Thus, by The Squeeze Theorem, 
.
You should point out that the inequality 
is also valid if 
is replaced by 
, since 
and 
. It is important to realize this because otherwise it could be argued that the proof that 
is incomplete since we only examined 
.
An important corollary needs to be addressed next.
Corollary: 
Proof: 
. Note the use of limit property 6 from C.5.
Another very important corollary needs to be developed.
Corollary: 
Proof: Let 
. Then as 
, 
. Thus 
.
Now it is time to return to one of the original limits, namely 
.
.
Thus 
. An immediate consequence is 
.
Summary of known trigonometric limits.
1. 
2. 
3. 
4. 
5. 
The following examples illustrate how the results above can be used to evaluate related limits.
Example 1: Find 
.
Solution: Look for a way to involve the corollary 
. Rewrite as shown: 
(result 3).
Example 2: Find 
.
Solution:
Example 3: Find 
.
Solution: 
Example 4: Find 
.
Solution: 
Example 5: Find 
.
Solution: 
Example 6: Find 
Solution: You cannot use 
since 
does not exist (see C.4 – oscillating discontinuities).
Now we know that 
. If 
, then we can multiply by x to obtain 
. As 
, both 
and x approach 0. Since 
lies between 
and x , it too, must approach 0 according to The Squeeze Theorem. Thus, 
. If 
, a very similar proof follows. Thus, 
.
The graph below shows how 
is squeezed between 
and 
near 
.
Examples/Activities
1. With a graphing calculator set to radians, draw the graphs of 
, 
, and 
in the window 
and 
. Which function is squeezed between the other two? Based on the graphs, what might be the value of 
.
2. With a graphing calculator set to radians, draw the graphs of 
, 
, and 
in the window 
and 
. Which function is squeezed between the other two? Based on the graphs, what might be the value of 
?
3. If 
, find 
and explain your reasoning.
4. If 
for 
, find 
.
It is important that students carry out the following activity. They will then have a more complete appreciation for the use of radian measure in section G.4. If degree measure was used instead of radian measure, the derivative of 
would not be 
, but would rather be 
– something we will see in section G.4.
Set your calculator to degrees, and complete the following table.
Based on your table, what is 
if 
is measured in degrees?
Based on your table, what is 
if 
is measured in degrees?
The assignment you create should contain a few limits that can be evaluated by direct substitution or by some other strategy different from the techniques introduced in this section.
Evaluate each of the following limits.
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 
11. 
12. 
13. 
14. 
15. 
(Hint: Let 
. Thus as 
, 
. Also 
. Thus, the question becomes 
.
But 
, since cosine and sine are co-functions.)
16. 
(Hint: Perform a sign analysis.)
17. 
18. 
Use The Squeeze Theorem and the method shown in example 6 to find each of the following limits.
1. 
2. 
3. 
4. 
(Hint: 
. Since exponential functions with bases larger than 1 are increasing functions, then 
. Now multiply each part of the inequality by 
, which is never negative, and obtain 
. Apply the Squeeze Theorem).
Suggestions/Extensions
Prove that 
by beginning with the same figure used on the previous page and observing that the length of segment PA is less than the length of the arc from P to B , which in turn is less than the length of segment QB .
If time constraints prohibit you from proving that 
, you will still need to establish the result for use in section G.4. You could provide a convincing argument (but not a proof) that this limit has a value of 1 by examining output values of 
for values of 
near 0 by using a calculator. Be sure the calculator is set to radians. In a similar manner, you can show that 
appears to approach 0 as 
approaches 0.
1. Find 
. (Hint: 
.)
2. Find 
.
Objective
G.4 Back to Top
To differentiate sine and cosine functions and solve related problems.
Instructional Notes
Having established that 
and 
in G.3, students are prepared to find the derivative of the sine function. When the derivative of the sine function has been developed from the definition of the derivative, the derivatives of the remaining five trigonometric functions is easily established.
It is wise to review the graphs of the sine and the cosine functions at the outset of the development so that the students have a clear picture of the curves for which to find the slopes of the tangent line. Review the amplitude, period length, domain, range, intercepts, and local extrema of the two functions.
for 
for 
The development of the derivative of the sine function could proceed as follows.
The function: 
Begin with the definition of the derivative: 
Apply the definition to the function: 
Direct substitution will not evaluate the limit. Expand 
using the formula for 
: 
.
Rewrite the quotient in an equivalent form – regroup, factor, separate.
As 
, the terms 
and 
remain the same, but the terms 
and 
move towards 0 and 1 respectively provided we are in radians – see G.3.
Having proven that if 
, 
, ask the students to determine the slope of the tangent line drawn to 
at specific x values such as 
, 
, 
, and 
. Be sure to examine the graph of the sine function to see that the resulting slopes are reasonable. Yes, students do need to know from memory the exact values of the trigonometric functions of all multiples of 
and 
.
Now that the derivative of 
has been established, generalize the result to find the derivative of 
where u is a function of x by using the chain rule.
If 
, then 
Before developing the derivative of the cosine function, students should be given the opportunity to differentiate a few functions involving the sine function. Some suggestions follow.
Example: Find 
if 
.
Solution: 
.
Example: Find 
if 
.
Solution: 
.
Example: Find 
if 
.
Solution:
Example: Find 
if 
.
Solution: Since 
is the product of two functions, the product rule will be needed. Thus 
. Simplifying, 
.
Example: Find 
if 
.
Solution: 
. Then 
.
Having developed the derivative of the sine function, ask the students how they might find the derivative of the cosine function without the use of limits . Students will need to recognize that because sine and the cosine are co-functions, it is possible to represent the cosine of any angle in terms of the sine of the complement of the angle – namely 
. In the course of the development students will also need to know that the sine of any angle is the same as the cosine of the complement of the angle, namely 
. Their progress should be along the following lines.
The function: 
Using knowledge of co-functions, rewrite 
: 
.
We know how to differentiate sine functions: 
.
Simplify and again apply knowledge of co-functions. 
Thus, if 
, 
.
Generalize this result to find the derivative of 
where u is a function of x .
If 
, 
With students leading, work through some derivatives involving cosine functions. Some suggestions are provided.
Example: Find 
if 
.
Solution: 
.
Example: Find 
if 
.
Solution: 
. Thus, 
Example: Find 
if 
.
Solution: Using the product rule, we have 
Example: Find 
if 
.
Solution:
When students are comfortable with differentiating sine and cosine functions, students can use this skill to analyze graphs and solve problems involving these functions. Three examples are included below.
Example: Consider the graph of the function 
for 
. Find the critical numbers and all local extrema.
Solution: If 
, then 
. Perform a sign analysis for 
. Have the students determine the signs of the exponential function and the cosine function.
The critical numbers are 
and 
since for these values 
. Notice that 
changes signs at these x values. Thus, 
or 
is a local maximum point since 
went 
, while 
or 
is a local minimum point since 
went 
.
Example: Two sides of a triangle have lengths of 7 cm and 8 cm. What is the size of the angle 
, included by these two sides, that will maximize the area of the triangle?
Solution: The area of a triangle can be expressed as 
(half the product of two sides with the sine of the angle included by the sides – Math C30). Applying that result to the triangle, we have
. Thus, 
. To determine the critical numbers, let 
. Thus, 
making 
. This is true if 
is any odd multiple of 
. Since 
is an angle in a triangle, it cannot be negative nor obtuse. Thus, 
. The area of the triangle is maximized for 
.
This example is only appropriate if you have focused on related rate problems in F.4, which were optional.
Example: A ladder of length 5 m leans against a vertical wall with its foot on level ground. If the foot of the ladder moves away from the wall at a rate of 1 m/min, how is the angle between the ladder and the ground changing when the foot of the ladder is 3 m away from the wall?
Solution:5, x , and 
can be related by the cosine ratio. Thus 
. Solving for x , we have 
. Note that the 5 carries the unit of metres. Since this is a related rate problem, differentiate relative to time, t .
We know that 
. We need to determine 
at the when moment. Clearly 
at the when moment.
Since 
, we can determine that 
. Substituting into 1 , we have:
What are the units of this answer? Is the answer in degrees/minute or radians/minute? Remember that the derivative of 
is 
only if we work in radians . Thus, our result is in radians/min.
The angle between the ladder and the ground is decreasing at a rate of 0.075 radians/minute or, if you prefer the answer in degrees/minute, make the conversion to 4.30 degrees /minute.
Examples/Activities
Ask the students what the derivative of the sine function would be if we worked in degrees, not radians. The limit process would proceed as with radians until you reached 
. Now from the activity suggested in G.3, we observed that in degrees, 
, and 
. Thus, the derivative of 
would be 
. Working in radians yields a more convenient answer.
1. What is the slope of the tangent line to the graph of 
at:
(a) 
(b) 
(c) 
2. Find the equation of the tangent line to the graph of 
drawn at:
(a) 
(b) 
(c) 
(d) 
3. Explain why in the first quadrant the graph of 
always lies below the graph of 
. (Hint: consider the slope of both functions at 
.)
Find the derivative of each of the following functions.
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 
22. 
23. 
24. 
25. 
Find the derivative of each of the following functions.
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 
Use implicit differentiation to find 
.
1. 
2. 
3. 
4. 
Find the local extrema, if any, for each of the following functions. Restrict the domain to 
. Confirm with a graphing calculator.
1. 
2. 
3. 
4. 
5. 
6. The function 
has many local extrema in the interval 
. Find the first such extrema. Confirm with a graphing calculator.
7.. The function 
has many local extrema in the interval 
. Find the first such extrema. Confirm with a graphing calculator.
8. The function 
has many local extrema in the interval 
. Find the first such extrema. Confirm with a graphing calculator. (Hint: Use 
to rewrite the derivative in a different form.)
Some samples of problems that can be solved using trigonometric functions follow. Caution: 4, 5, and 6 are related rate problems – optional topic.
1. The height of a Ferris wheel rider above the ground after t seconds on the ride is given by the function 
.
(a) What is the rider's height above ground after 5 seconds?
(b) Find the vertical velocity of the rider at any time t .
(c) What is the vertical velocity of the rider when 
?
(d) When does the rider first reach a maximum vertical velocity?
2. All triangles inscribed in a semicircle are right triangles. Find the size of 
that will maximize the area of the triangle if it is inscribed in a semicircle of radius 20 cm.
3. What is the length of the shortest ladder that will reach across the vertical wall of height 2m and still reach the building if the wall is 3 m from the building? (Hint: Express the length of the ladder, namely 
, in terms of 
.)
4. (Optional) A kite is flying on the end of a string 130 m in length. The other end of the string is attached to a stake in the ground. A gust of wind causes the height of the kite to increase at a rate of 3 m/s. At what rate is the angle of elevation of the kite changing when the kite is 120 m high?
5. (Optional) A pair of shoes is attached to a cord and swung around in a circle “lasso style”. The rate of rotation of the shoes is increasing at a rate of 0.25 radians per second and, at the same time, the length of the cord is being let out at a rate of 0.5 metres per second. Find the linear speed of the shoes when the shoes are rotating at a rate of 2 radians per second and the cord is 4 metres long. (Hint: Recall that 
. Solve for s , and then use the product rule to find 
.)
6. (Optional) A door, 90 cm in width, is closing against a wall so that the angle between the door and the wall is decreasing at a rate of 
per second. At what rate is h , the distance between the end of the door and the wall changing, when 
and 
cm?
7. The sides of a V-shaped trough meet at an angle of 
. Find the size of 
that will maximize the cross-sectional area of the trough.
Suggestions/Extensions
It can be shown that 
if h is in degrees. Note that 
. Thus in degrees, the derivative of 
could be written accurately as 
.
Discuss with the students why radians provide more convenient situations than degrees by reminding students that degrees are based on a randomly chosen value (180° in a straight angle or 360° in a circle) while radians are defined in terms of the actual circumference of a unit circle.
Have the students create and share examples of transcendental functions that do and do not require the use of the chain rule when being differentiated in order to prove recognition of how functions are defined and composed.
Have the students graph 
and 
and discuss what they understand from the graph and from the meaning of a derivative. Repeat with 
and 
.
Using a trigonometric function approach, find the dimensions of the largest rectangle that can be inscribed in a semicircle of radius 10 cm. Solve the problem a second time, without using trigonometric functions. Compare the two solutions in terms of complexity and number of steps.
This would be an opportune time for students to research hyperbolic functions (IL). Certain combinations of exponential functions occur so often in mathematics that they are given special names. Two of them are defined below. 
You pronounce 
as “shine” or “cinch” while 
is pronounced as “Kosh” rhyming with the “kosh” in kosher or rhyming with “gosh”. Some people prefer to refer to them as hyperbolic sine and hyperbolic cos. These functions are related to the hyperbola 
in the same way that 
and 
are related to the unit circle 
.
Hyperbolic functions have many of the characteristics of trigonometric functions.
Show that:
• The derivative of 
is 
.
• The derivative of 
is 
.
Hyperbolic functions describe the shapes of hanging power lines, wave motion in elastic solids, and temperature distributions in cooling fins. The arch in St. Louis , Missouri is in the shape of an upside down hyperbolic cosine function – namely a 
function.
See Appendix A for additional objectives and activities.
