Differentiation basics

teaching materials

The introductory content for differential calculus is similar for SL and HL students; however, SL students do not need to learn the limit definition of the derivative (also known as finding a derivative from first principles).

Although I prefer an emphasis on concepts - such as rates of change, limits of functions, relationship between gradient (slope) of a secant line and slope of a tangent line, slope of a curve at a point, etc - I also think it's important that students develop confidence with the rules of finding derivatives (differentiation).

Teaching Notes

A very important early concept in the study of differential calculus is the rate of change of a function. Students are very familiar with this idea in the context of a linear function - that is, the slope (or gradient) of a line. But the rate of change (slope) of a linear function is constant. It's critical to get students thinking about how to describe the slope of a function that is not linear. Since the slope is changing the question changes to how to describe/define the slope of a curve at a specific point. An effective approach is to discuss with students the difference between average velocity and instantaneous velocity. Present students with some examples of distance-time graphs from which they can compute the average velocity for an indicated time interval and also make an educated guess about the velocity at a specific single moment (at a single point on the graph rather than over an interval with two endpoints).

Figures 1 & 2 show the graph of a function that gives the distance (d meters) of an object at a particular time (t seconds). Velocity is the rate of change of distance per unit of time. Figure 1 shows both graphically and numerically the average velocity for the object from 4 sec to 10 sec. Graphically this is represented by a line that intersects the function at two points - the endpoints of the time interval. A secant line is a line that intersects a graph at two points. Figure 2 shows a graphical representation of the velocity at the specific time of 7 sec - that is, the instantaneous velocity at t=7. This instantaneous rate of change is the slope of a tangent line. With some dynamic (and preferably interactive) animations students quickly grasp the idea that the tangent line can be approximated by a secant line with the approximation improving as the secant line's two points of intersection get closer to one another.

A key idea to get across to students (SL & HL) is that this physical process of a secant line getting closer and closer to a tangent line represents a limiting process. As the difference between the two intersection points of the secant line approaches zero the slope of the secant line approaches the slope of the tangent line - i.e. the rate of change of the function at the point of tangency. We now will call this the derivative of a function at a point. Even though SL students are not required to find a derivative from first principles (applying limit definition of the derivative of a function), this development will help them understand why we define - in a geometric/visual sense - that the value of the derivative (rate of change) of a function at a particular point is equal to the gradient of the line that is tangent to the graph of the function at that point.

HL students: Before the derivative can be defined in a more precise mathematical manner it is important for HL students to gain an informal understanding of the limit of a function along with basic properties of limits. Once students have done this then they are ready to develop the formal defintion of the derivative of a function at a point as the limit of the slope of a secant line as the difference in the x-coordinates of the secant line's two points of intersection goes to zero. I think it is very helpful for HL students to be able to express this definition in words and not just only be able to copy down the definition from the formula booklet.

There are numerous websites that show animations of the definition of the derivative. If you find one you like it is certainly worthwile to consider using / sharing it with your students. Whenever possible I like my HL students to create their own dynamic illustrations, if possible - considering type and availability of software. My preferred software for composing graphical / geometric dynamic illustrations is Geometers Sketchpad. It can be used for a student (or teacher for classroom demonstration purposes) to interact with the dynamic illustration of this very important idea at the start of the study of calculus.

Geogebra appletsecant-tangent-derivative

The breadth and depth of material to be covered for differentiation basics is fairly close to the same for SL and HL. HL will need some practice with the process of finding a derivative from 'first principles'. Use of this abstract process of finding a derivative by applying the limit definition of the derivative (first principles) will only be for relatively low-degree polynomial functions. It is not a common exam question. SL students often find it a bit easier to understand the correspondence between the rate of change of a function at a point and the slope of the line tangent to the function at that point by spending sufficient time looking at examples that involve some kind of real-life context. The simplest examples - and ones find more accessible - are ones involving distance-time graphs.

 ♦ teaching materials

Differentiation_practice_1_v1   smiley laugh  
Differentiation Practice-1:  Exercise set with 7 questions (some are for HL only).  Syllabus content covered is listed at start.  No GDC allowed.  Worked solutions are included. This exercise set is the 1st in a set of 4 intended to give students practice in many of the basic techniques and applications of differential calculus.

Differentiation_practice_2_v1  smiley laugh
Differentiation Practice-2: topics covered include differentiation of polynomials; chain rule; quotient rule; identifying and verifying stationary points and inflexion points; finding lines tangent and/or normal to a curve. This is the 2nd exercise set in a set of 4 that cover many of the basic techniques & applications of differential calculus. Worked Solutions are included

Differentiation_practice_3_v2   smiley laugh
Differentiation Practice-3:  Exercise set with 15 questions (answers included).  No GDC allowed.  This exercise set is the 3rd in a set of 4 intended to give students practice in many of the basic techniques and applications of differential calculus. [v2: corrected coordinates for point of tangency in Q.6]

Differentiation_practice_4_v1  smiley laugh
Differentiation Practice-4: Exercise set with 13 questions. No GDC, except on three of the questions. This is the 4th and last exercise set for covering many of the basic techniques & applications of differential calculus. Worked solutions available below.

Differentiation_practice_4_SOL_KEY_v1  smiley laugh
Worked solutions for the Differentiation-4 exercise set above.

All materials on this website are for the exclusive use of teachers and students at subscribing schools for the period of their subscription. Any unauthorised copying or posting of materials on other websites is an infringement of our copyright and could result in your account being blocked and legal action being taken against you.