Mental Constructions Involved in Differentiating a Function to a Function Power

Rachel Rupnow Catherine Ulrich Virginia Tech Virginia Tech

Functions of the form 𝑓 𝑥 = (𝑔 𝑥 )!(!), including constant functions, power functions, and exponential functions, are fundamental examples of functions that differential calculus students should be able to differentiate. Yet students often struggle to distinguish between these forms. Drawing on APOS (Action-Process-Object-Schema) theory as well as Piaget and Garcia’s triad of schema development, this paper offers a genetic decomposition of the schemas students build for determining the derivative of a function to a function power. In particular, we analyze how students determine which differentiation rules to use with different function structures of a function to a function power and how students construct a conception of logarithmic differentiation. An initial genetic decomposition informed by existing literature was refined using the results of a series of three clinical interviews with each of two calculus students. Findings include the necessity of a strong background in functions, logarithms, and other differentiation rules. Key words: Logarithms, Differentiation, Function, APOS, Schema

Purpose and Background

Logarithmic differentiation is useful when differentiating non-constant functions to non-constant powers. For example, consider the function 𝑦 = 𝑥!. To find the derivative of this function, a student would need to realize that neither the power rule nor the exponential rule applies so the function equation must be transformed into an equation to which standard differentiation rules apply. Taking the natural logarithm of both sides of the equation and implementing a property of logarithms yields ln 𝑦 = ln 𝑥! = 𝑥  ln 𝑥. This function can then be differentiated using standard differentiation rules, resulting in !"

!"  =  𝑥! ln 𝑥 + 1 . When either the

base or exponential function in a function to a function power (FFP), 𝑓 𝑥 = (𝑔 𝑥 )!(!), is constant you can, in contrast, use the appropriate rule for differentiating polynomial or exponential functions.

Therefore, appropriately carrying out logarithmic differentiation requires students to coordinate all other differentiation rules and properties of logarithms. Additionally, students who know when to apply the technique demonstrate recognition of differences between types of FFPs. Because of this, we felt that differentiating FFPs provided a rich context for studying how students construct and utilize their differentiation rules. In this paper, we examine the necessary mental constructions for students to differentiate FFP expressions and to distinguish between situations where the constant rule, power rule, exponential rule, and logarithmic differentiation are appropriate.

Theoretical Framework

We use both APOS theory (Dubinsky, 1991) and Piaget and Garcia’s (1989) triad of schema development to analyze the mental constructions necessary to address problems like the one above. APOS theory is an extension of Piaget’s work with reflective abstraction in which student

concepts are categorized as Actions, Processes, Objects, or Schemas. An action is an external transformation of objects (Asiala, et al., 1996); for example, a student with an action concept of the derivative could directly apply the power rule to find the derivative of 𝑓 𝑥 =  𝑥!. A process is an interiorized action; for example, a student with a process conception of the derivative could find the derivative of 𝑓 𝑥 = 𝑥! + 2 ! by expanding the binomial and then taking the derivative by using the power rule or by applying the chain rule and power rule. An object is an encapsulated process; for example, a student can think about the result of applying the power rule to an appropriate, arbitrary function as producing another function, not just as a series of steps to find a solution. A schema coordinates a student’s objects and processes; for example, a student could recognize that finding the derivative of 𝑓 𝑥 = 𝑥! + 2 ! could be accomplished by expanding the binomial and then taking the derivative by using the power rule and could be accomplished by applying the chain rule and power rule, but then decide which was simpler and differentiate accordingly.

Researchers such as Clark et al. (1997) have found categorization using APOS theory difficult in cases where students will be at different levels of abstraction for different elements of study: action concepts of some composite functions and process for others when studying the chain rule. In these cases, the triad of schema development can help look in more detail at how schemas develop (Dubinsky & McDonald, 2001). Three stages, Intra, Inter, and Trans, describe the development and coherence of connections made within a student’s schema. At the Intra stage, a student focuses on individual objects rather than looking for connections between them. A student at the Inter stage recognizes some relationships between different actions, processes, objects, and schemas, but cannot explicitly connect all of the relationships. A student at the Trans stage has created a coherent structure that connects appropriate relationships and recognizes what is and is not within the scope of a schema.

Preliminary Genetic Decomposition

Students with only an action conception of function need to substitute specific values into a function and receive outputs to make sense of a function. Students with a process conception of function recognize that a function receives inputs and gives outputs without explicitly needing values with which to calculate, but still consider a function in terms of a dynamic activity (Arnon et al., 2014). A student with an object conception of function recognizes the set of outputs of a function as an entity, recognizes the relationship of the inputs to the outputs, and can distinguish between different types of functions. We hypothesize that an object conception is necessary when differentiating FFPs in order to distinguish between types of functions and choose applicable differentiation rules and techniques.

An action conception of logarithms requires logarithms to be calculated with specific values to have any meaning. At the process level, students can recognize logarithms are functions that obey specific properties and use those properties of logarithms appropriately because they recognize the process that has been applied to compute solutions. For many students, it takes a long time to attain a process understanding of logarithms, as they tend to overgeneralize algebraic rules as they “factor out” logarithms (i.e., ln 𝑥 + ln 𝑦 = ln(𝑥 + 𝑦)) or “cancel” logarithms (i.e., !"!

!"!=   !

!), (Liang and Wood, 2005). This process knowledge is necessary for

logarithmic differentiation because complex equations involving the chain rule, such as 𝑓 𝑥 =   (𝑥! + 2)!!!, require awareness of what can and cannot be separated using logarithmic properties.

A differentiation scheme at any level of development, from action conception to full schema development, includes many subschemes including a graphing scheme and separate schemes for determining derivatives according to different differentiation rules. In order to address finding the derivative of an FFP, all differentiation rules must be at least at a process level for derivatives requiring more than one rule to be coordinated in a new process, much as Maharaj’s 2013 study concluded. However, each differentiation rule need not be viewed as a static object, because coordinating the rules and completing the procedure for finding the derivative could be viewed as a few more steps in a dynamic process.

Finally, students must develop a logarithmic differentiation scheme as a subscheme of their differentiation schemes. The logarithmic differentiation scheme coordinates their function scheme, logarithm scheme, and various differentiation rule subschemes in order to determine a derivative of an FFP when the functions in both the base and exponent are non-constant. Because this scheme must coordinate so many other schemes, decomposing students’ mental constructions with the triad of schema development is useful.

Students at the Intra level of logarithmic differentiation would struggle with coordinating the constituent schemes described above, either because they had not developed all of them to a sufficient degree or because they would not recognize that their other differentiation schemes could not be applied to a situation. For example, students may not recall their logarithmic properties and therefore take the natural logarithm of the equation, but then use the power rule to differentiate the resulting logarithm rather than applying properties of logarithms, which was the point of performing logarithmic differentiation (i.e. 𝑦 =  𝑥!  yields ln 𝑦 = ln 𝑥!, but then the student incorrectly concludes !

!!"!"=   !

!!!

!!). Alternatively, students might be able to perform the

steps of logarithmic differentiation but would need to be told to apply it. Another possibility is that they might indiscriminately perform the process to find the derivative of any function, whether useful or not (i.e. determining the derivative of 𝑦 = sin!! 𝑥 through logarithmic differentiation because it appears to contain a power).

Students at the Inter level would have organized the constituent differentiation rule subschemes sufficiently to recognize which functions require logarithmic differentiation. However, they would not recognize why logarithmic differentiation works or why functions with constant base and non-constant power and functions with non-constant base and constant power require different differentiation rules or why logarithmic differentiation might be applied to equations like  𝑓 𝑥 =  𝑥 sin 2𝑥3 cos 3𝑥2 + 2 .

Students at the Trans level would recognize that logarithmic differentiation is useful because it allows previously inaccessible functions to be transformed into functions that can be addressed with other differentiation rules. A secondary use of logarithmic differentiation, simplifying functions requiring multi-part product rule application like 𝑓 𝑥 =  𝑥 sin 2𝑥! cos 3𝑥! + 2 , would also be assimilated at this time, though a student may not choose to use logarithmic differentiation in this way. Moreover, students might recognize that changing the position of the constant in FFP situations (i.e. 𝑦 = 𝑒! versus 𝑦 = 𝑥!) alters the set of outputs of the function, so it also alters the rate of change of the function.

Methods

The participants for this study were two entering freshmen taking a differential calculus course in a six-week summer session. Both students had previously seen differential calculus in

high school, participant A through AP Calculus AB and participant B through a pre-calculus course that also included calculus topics. Participants were recruited from the same section.

Each participant engaged in three semi-structured interviews (Fylan, 2005) lasting 30-50 minutes. The interview questions are listed in Figure 1. The first interview occurred after both students had been taught how to find derivatives of constants, power functions, 𝑒!, and to use the product, quotient, and chain rules. The second interview occurred after students had been taught logarithmic differentiation. The last interview occurred within a day of the final exam for the course, approximately three weeks after the second interview. The interviews were video-recorded and participants’ written work was collected. To analyze the data, participants’ interviews were examined for evidence of the construction of processes, objects, and schemas relevant to developing a schema for logarithmic differentiation.

Interview 1 Interview 2 Interview 3 𝑓 𝑥 =  𝑒! + 3𝑥 𝑓 𝑥 =  5 sin(𝑥!) + 𝑥𝑒! Sketch  the  graph  of  the  derivative  of  the  given  

function  (which  was  𝑓 𝑥 = 2.82𝑥 + 1.37). 𝑓 𝑥 =  4𝑥! − 2𝑒! 𝑓 𝑥 =  ln  (2𝑥! − 1) 𝑓 𝑥 =  𝑥! 𝑓 𝑥 =  4!! − cos! 𝑥 Sketch  the  graph  of  the  derivative  of  the  given  

function  (𝑓 𝑥 = 2.9 𝑥 + 2.6 ! − 2.1). 𝑓 𝑥 =   𝑥! + 1 ! 𝑓 𝑥 =  𝑥!

𝑓 𝑥 =  7𝑒!!

𝑒! 𝑓 𝑥 =  

cos 𝑥𝑥

Sketch  the  graph  of  the  derivative  of  the  given  

function  (𝑓 𝑥 = 𝑒!! + 2). 𝑓 𝑥 =  𝑒! Sketch  the  graph  of  the  derivative  of  the  given  

function  (𝑓 𝑥 = 3𝑥! + 2𝑥! + 1). 𝑓 𝑥 =  𝑥𝑒! 𝑓 𝑥 =  𝑥 sin 2𝑥! cos 3𝑥! + 2

𝑓 𝑥 =  𝜋𝑥! Sketch  the  graph  of  𝑓 𝑥 = 𝑥!  and  connect  to  the  derivative  of  the  function.  (The  derivative  is  

𝑓! 𝑥 = 1 + ln 𝑥 𝑥! .) 𝑓 𝑥 =   (𝑥! + 2)!!!   𝑓 𝑥 = (𝑥! + 2)!!!

𝑓 𝑥 = (𝑥! + 4) sin 𝑥   𝑓 𝑥 =   sin!! 𝑒 !!! ! + 𝑒!.! Differentiate:  𝑓 𝑥 =  𝜋! + 2! + 𝑥! + 𝑥!/! Figure 1: Interview questions

Results

Both participants demonstrated at least an object conception of function; they could fluently distinguish between constant, 𝑥!, and 𝑛! forms when asked to identify what type of function each was. Participant A was also able to identify graphs of quadratic and cubic polynomials and exponential functions, though Participant B struggled to identify the graph of an exponential function. Additionally, when Participant A was asked to find the derivative of 𝑓 𝑥 =  𝑥! in the first interview, before learning logarithmic differentiation, he noted similarities to 𝑓 𝑥 =  𝑒!, but also noted the base 𝑥, “was not a constant, or a number. It’s just weird.”

Participant B displayed a process conception of logarithms. He applied logarithm properties appropriately in all but one problem and actively avoided the error he had made in the second interview in the last interview. Specifically, in the second interview he chose to use logarithmic differentiation on 𝑓 𝑥 =  4!! − cos! 𝑥, which led him to say that ln 𝑦 = 2𝑥 ln 4 − 3 ln (cos 𝑥). However, in the last interview, he recognized directly applying logarithmic differentiation to 𝑓 𝑥 =  𝜋! + 2! + 𝑥! + 𝑥!/! “wouldn’t help because it’s not multiply or divide; it’s the sum. They are different [functions] so that’s not going to work.”

Participant A was less comfortable with the use of logarithms, displaying process level conceptions intermittently. In interview 2, when asked to differentiate 𝑓 𝑥 =  ln  (2𝑥! − 1), he questioned aloud whether or not 𝑓 𝑥 = ln 2𝑥! − 1 = ln 2𝑥! − ln  (1). When asked if he could determine if this was true or not, the only method he could determine was to compare their derivatives. While this method worked, he could not construct an argument based on properties

of exponents. In interview 3, when asked to differentiate 𝑓 𝑥 =  𝜋! + 2! + 𝑥! + 𝑥!/!, he took the natural logarithm of both sides, but wrote each piece separately as ln 𝑦 = ln 𝜋! + 𝑥 ln 2 + 2 ln 𝑥 +!!ln 𝑥. While he knew some procedures for working with logarithms, he did not display

understanding of why the procedures worked. Both participants had full schema conception for all of their differentiation rule subschemas.

They were both very comfortable with differentiating using the product, quotient, chain, and basic differentiation rules. In many instances, they demonstrated multiple ways of approaching problems, including recognizing the derivative of 𝑓 𝑥 = 𝑥! + 1 ! could be found by using the chain rule directly or by expanding the binomial and then applying the power rule.

Bringing together these constituent subschemas, Participant A had an Inter stage and Participant B had a Trans stage of logarithmic differentiation schema development. By the end of the course, neither participant tried to apply his logarithmic differentiation schema to functions where it was not helpful (like 𝑓 𝑥 =   sin!! 𝑒 !!! ! + 𝑒!.!) and both recognized when it was necessary (𝑓 𝑥 =  𝑥!) or helpful (𝑓 𝑥 =  𝑥 sin 2𝑥! cos 3𝑥! + 2 ). However, in interview 3, Participant A could not successfully differentiate 𝑓 𝑥 =  𝜋! + 2! + 𝑥! + 𝑥!/! because his logarithm schema was not sufficiently developed, whereas Participant B could and realized why his previous “factoring” of the logarithm was inaccurate.

Discussion

Because both participants struggled at times to process how a logarithm works and what properties it possesses, having a process conception of logarithms does appear to be necessary to address logarithmic differentiation problems. Despite expecting development of a Trans stage schema for logarithmic differentiation to be necessary for students to consider apply logarithmic differentiation to three-part product rule situations, Participant A, who only attained Inter stage, applied this ably. Thus we revise the genetic decomposition, claiming students recognize all possible opportunities to use logarithmic differentiation at the Inter stage, even if they do not understand exactly why it works.

The present genetic decomposition of a logarithmic differentiation schema requires an object level conception of function and a process level conception of logarithms and each differentiation rule. A student’s logarithmic differentiation schema then coordinates these subschemas. The level to which a student is able to coordinate these subschemas is characterized as being at the Intra level if the student cannot coordinate them and does not realize when logarithmic differentiation should be performed, at the Inter level if the student recognizes when to apply logarithmic differentiation but does not understand why, and at the Trans level if the student recognizes the underlying structure of the functions and can therefore answer why logarithmic differentiation should be used.

By describing the mental constructions necessary to perform and know when to apply logarithmic differentiation through APOS theory and the triad of schema development, instructors may be better equipped to teach in a manner that encourages the development of students’ mental constructions. Specifically, encouraging students to distinguish between different types of functions may assist students in strengthening their function schemas and differentiation rule schemas. This will then enable them to construct a logarithmic differentiation schema and a richer understanding of calculus.

References

Arnon, I., Cottrill, J., Dubinsky, E., Oktaç, A., Roa Fuentes, S., Trigueros, M., & Weller, K. (2014). Genetic decomposition. In APOS theory: A framework for research and curriculum development in mathematics education, pp. 27–55. New York, NY: Springer.

Asiala, M., Brown, A., DeVries, D. J., Dubinsky, E., Mathews, D., & Thomas, K. (1996). A framework for research and curriculum development in undergraduate mathematics education. Research in Collegiate Mathematics Education II, CBMS Issues in Mathematics Education, 6, 1–32.

Clark, J., Cordero F., Cottrill, J., Czarnocha, B., DeVries, D. J., St. John, D.,…, Vidakovic, D. (1997). Constructing a schema: The case of the chain rule. Journal of Mathematical Behavior, 16(4), 345–364.

Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advanced mathematical thinking, pp. 95–123. Kluwer: Dordrecht.

Dubinsky, E. & McDonald, M. A. (2001) APOS: A constructivist theory of learning in undergraduate mathematics education research. In D. Holton (Ed.), The teaching and learning of mathematics at the university level: An ICMI study, pp. 275–282. Kluwer: Dordrecht.

Fylan, F. (2005). Semi-structured interviewing. In J. Miles and P. Gilbert (Eds.), A handbook of research methods for clinical and health psychology (pp. 65–78), Oxford: Oxford University Press.

Liang, C. B. & Wood, E. (2005). Working with logarithms: Students’ misconceptions and errors. The Mathematics Educator, 8(2), 53–70.

Maharaj, A. (2013). An APOS analysis of natural science students’ understanding of derivatives. South African Journal of Education, 33(1), 1–19.

Piaget, J., & Garcia, R. (1989). Psychogenesis and the history of science (H. Feider, Trans.). New York: Columbia University Press.