# the limit does not exist

##### or a deep dive into differentiation from first principles

One of the changes made with the introduction of the new A level in Mathematics in 2017 was the additional requirement of knowing and applying differentiation from first principles. This gives students a fascinating insight into the world of university level calculus and should be celebrated. However it certainly introduced a new level of challenge into the A level qualification: it is new notation, new concepts and new procedures.

In this post I will explore differentiation from first principles within the A level and discuss how I teach it. This is hugely inspired by Jo Morgan and her excellent Topics in Depth series.

###### spec and assessment

First the context. Below is the specific content from the specification. This is taken from Edexcel, but the other exam boards are very similar.

AS level students need to be able to prove, using differentiation from first principles, the derivative of $x^n$ for $n=1, 2, 3$. For the full A level students need to also do the same for sin and cos. The formula is given in the formula book in function notation:

In terms of the assessments the table below shows when the topic has come up so far for Edexcel:

It appeared in both levels in 2018, but not at all in 2019. Interestingly though, the full A level assessment in June 2019 did include the limit definition of the integral (Paper 2 Q5). The specific questions in the assessments were:

For the first question the examiners report is relatively positive. But it is mentioned that several students did not understand what the question was asking and simply stated the derivative or the general rule. Some students lost the final marks for either incorrect notation, incorrect algebra or for not fully connecting the idea of a chord with the gradient.

The report for the second question again is relatively positive, with most students making a correct start. It is mentioned that some students incorrectly used the limit or did not use correct notation.

###### what does it lead to?

The limit definition of integration within the A level uses a similar idea, although for that students only need to recognise the notation involved rather than directly prove results. Students taking the Further Maths A level will explore limits more and see similar ideas in L’Hôpital’s rule. First year maths degrees will cover limits and the formal definition of differentiation as major topics. This topic is the first real introduction students have to the world of mathematical analysis and formal calculus.

###### how i teach it

This would be my sequence of learning within the year 12 topic of differentiation. It doesn’t necessarily correspond to lessons, just the order of topics. This is based on the idea of atomising – breaking complex methods into their constituent sub-steps.

1. differentiating $x^n$ (with no mention of graphs or gradients yet, just the process)
2. differentiating expressions like $ax^n$, then ones like $ax^n + bx^m + \dots$
3. using differentiation to find the gradient at a point (now emphasising the graphical representation)
4. using the gradient to categorise increasing, decreasing and stationary
5. finding tangent lines
6. finding normal lines
7. limit notation
8. differentiation by first principles
9. finding second derivatives
10. etc

I first introduce differentiation as a process to follow and ensure students can differentiate things like $4x^2 - 2x +1$ before I talk about gradients. If you try introduce gradients and graphs at the same time I find it becomes overloading.

Above is example 1 in the differentiation chapter of the Oxford Edexcel A level textbook. As in, this is the very first example students see in the differentiation chapter. Many other textbooks put differentiation by first principles right at the start of the chapter too. Most A level students will not have seen differentiation at all before. Trying to introduce that many concepts at once will not be successful. This is why I make sure students are confident differentiating and knowing what it means graphically before I introduce first principles.

Part 7 in the sequence of learning is limit notation. I think it is essential that students see and experience limit notation before you include it within differentiation by first principles and that this is the step most likely to be overlooked. It’s atomisation again. I would introduce this notation, explaining that it is read as ‘the limit as h tends to zero of f of h‘,

I would explain to students that this means ‘what happens to the value of the function f as h gets closer and closer to zero’. I would give students several examples. Notation is important here so I would emphasise how we write it, consistently vocalising ‘the limit as h tends to zero of… is equal to…’ and ensure they are writing it correctly. The examples I cover might look something like this.

Then I would get the students to work through a variation style series of questions such as this.

Notice that there are non-examples here in n) and v). Questions e) and m) may require some prompting – but ask them to look at the previous questions. The most important take-away is that if students can do e), m), t) and u) then they will feel secure with limits when it comes to differentiation from first principles.

Students only need to know the limit as h tends to 0, but for some additional challenge and exploration you could include limits approaching other numbers like,

So far this practice with limits is completely disconnected to differentiation – and that’s deliberate. In my experience this only takes half an hour or so. This investment means that students are then better prepared for differentiation by first principles.

Then it’s onto part 8 – actually teaching differentiation by first principles. I would start by showing them the formula. I would tell them that we already know how to find the derivative but this formula is a way to prove it, emphasising that this formula is in the formula book. Again, it’s the consistent use of ‘the limit as h tends to zero of…’,

I would then use $f(x)=x$ as a first example. I would explain to students that we already know this differentiates to 1 but we are going to prove it. I would work through it like this, again emphasising notation and vocalising each new line as ‘the limit as h tends to zero of… is equal to …’,

Because they have already done d) and e) in the limit questions above, and they already know what limits are, this process is a lot more approachable – the only new part is substituting into the definition. I would go through one or two more examples and then let the students work through several examples themselves building in difficulty. Something like $2x$, $x^2$, $3x^2$, $2x + 1$, $3x -1$, $x^3$ etc. Students know at this point how to differentiate these so they know what answer to expect.

Students really must be fluent with algebraic manipulation here. Their numerators will involve expressions like $3(x+h)^2 - 3x^2$. I’ve seen a student write that that’s equal to $h^2$ ‘because the $3x^2$s cancel’.

I’ve found it is important here to check students’ work regularly. They must be using correct notation and I make them layout their answers like mine above. I mark their work myself or peer mark it to ensure they are using the notation correctly. I include examples with more than one term, like $2x+1$, because it did appear on a specimen paper.

There are other correct ways to answer these questions, considering chords, taking the limits at a later step or using $\delta x$. But sticking with the formula and h from the outset I find to be the most efficient and least likely for students to make mistakes with.

After they are comfortable with the algebra I would then show the graphical representation. I would argue it is important to do this after so you don’t overload students. The what before the why.

• The green line is the tangent at the black point.
• The blue line is the straight line through the black point and the blue point.
• The blue point moves closer (h gets smaller) and the blue line approaches the green tangent line.
• The gradient of the blue line is the expression within the limit – it tends to the gradient of the green line (the tangent) as h tends to zero.

I would let students play with this and explore what is happening. I would argue that they can appreciate this diagram more after they have learnt to find tangent lines and practised differentiating from first principles.

This post is already lengthy – so I won’t cover differentiating trig by first principles here.

###### misconceptions

I have found there are two big misconceptions with this topic. Over-application and notation.

After I first introduced this to students I then kept seeing students use this process when they didn’t need to. They would be using first principles with everything they saw. It is important to tell students that they only do this when the question asks from first principles. From first principles means prove it, prove it when it says first principles. Otherwise use the standard results.

The second is notation, and this picks up with the examiners report mentioned earlier. I’ve seen students write things like this:

Which is not correct, it’s only true with the limit. You can use the graph above to show students that it’s not correct – the green and blue lines are not equal for any h, you need to take the limit. I’ve seen students write things like this

Which again is not true, it’s only true in the limit. I’ve seen students write things like this

Which is an incorrect use of notation. This is why I emphasise throughout that students must follow my structure, why I check students workings and why I consistently say ‘the limit as h tends to zero of … equals…’. As they are becoming confident I will show students these misconceptions and get them to explain what is wrong with them. Bad understanding of notation is a reason students drop marks.

As I mentioned earlier, another issue is that students will not apply differentiation from first principles when it is needed in the exam, and will just quote the general rule. This is countered by hammering ‘first principles means prove it, prove it when it says first principles’.

Algebraic weaknesses can also be an issue. Students must be fluent expanding and simplifying expressions like $3(x+h)^2 - 3x^2$ and if not that should be tackled first.

One final mistake I saw a lot is disappearing negative signs. If your function is $f(x)=x-2$ then when students write out the limit they’ll have

I’ve found it is common to see the negatives in the highlighted part go astray.

Differentiation from first principles is a fantastically rich topic that gives students a great introduction to university level calculus. It is a great joy that it is now on the A level course and genuinely one of my favourite topics at A level. But because it is a new topic with new concepts and new notion it must be taught with care and sequenced effectively to ensure students understand it well.

The questions used in this post can be downloaded from this powerpoint.