IB Maths: Differentiation From First Principles (AA)

You need the limit definition $f^{\prime }(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$, plus the ability to apply it to simple polynomials such as $x^2$ or $3x^2-x$. On the AA syllabus this sits in topic 5.1, tested with 'show that' or 'find, from first principles' questions worth roughly four to six marks.

The formula is $f^{\prime }(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$ — the gradient of a chord between two points on the curve as the gap $h$ shrinks toward zero. It does not appear in the exam formula booklet, so it's one of the few AA definitions you genuinely need memorised, not looked up.

A limit is the value a function approaches as its input gets arbitrarily close to a point, without necessarily reaching it. In first principles differentiation, you find the limit of a chord's gradient as $h\to 0$, converting an average rate of change into the instantaneous gradient at one exact point.

Because examiners test understanding, not just mechanics. A 'differentiate from first principles' question explicitly forbids the power rule — using it skips the required method and loses marks. It also proves why the power rule works at all, which underpins Paper 1's 'show that' style questions.

It almost always appears on Paper 1, the non-calculator paper, since it's an algebraic limit process rather than a numerical one. Expect a short, structured question worth around four to six marks, often early in the paper, asking you to differentiate a simple polynomial like $x^2+2x$.

It's core content for both SL and HL Maths AA — not an HL-only extension. HL students additionally meet first principles proofs for $\sin x$, and occasionally $e^x$, using compound angle or exponential limit identities that SL students never need to reproduce.

Usually 'find' or 'show that'. 'Show that' gives you the answer already, so every limit step must appear on the page to earn method marks. 'Find' means you derive the result yourself, with marks awarded for correctly setting up the limit expression before you even simplify.

Substitute $f(x)=x^2$ into the limit definition, expand $(x+h{)}^2$, cancel the $x^2$ terms, divide every remaining term by $h$, then let $h\to 0$. The surviving term is $2x$, so $f^{\prime }(x)=2x$ — matching the power rule, which is exactly the point of the exercise.

This is HL content. Use the compound angle identity $\sin (x+h)=\sin x\cos h+\cos x\sin h$, then split the resulting limit using the two standard results ${\lim }_{h\to 0}\frac{\sin h}{h}=1$ and ${\lim }_{h\to 0}\frac{\cos h-1}{h}=0$. Everything collapses neatly to $f^{\prime }(x)=\cos x$.

The most common one I see marking mocks: substituting $h=0$ before simplifying, which gives $\frac{0}{0}$ and stalls the whole answer. Others forget to factor $h$ out of the numerator, or write $f(x+h)$ as $f(x)+h$ — a notation slip that quietly costs the entire method mark.

There's no real comparison — Mathematics: applications and interpretation doesn't include first principles differentiation at all; it's an AA-only topic at both SL and HL. If your child is choosing between the two subjects, this is one of several proof-style, purely algebraic topics that live in AA rather than AI.

On its own, not much — typically one short question worth four to six marks out of well over a hundred available across both papers. But it signals whether a student understands calculus conceptually, and that understanding pays off later in optimisation and related-rates questions, where marks add up fast.

Practise the algebra until the cancellation step is automatic, not just the final answer. Work through past-paper 'show that' questions under timed conditions — the structure (expand, cancel, divide by h, take the limit) repeats almost unchanged from session to session, so pattern recognition beats memorising individual proofs.

Yes — the IB's own subject guide and specimen papers are the primary free source, with full past papers available through a school's IBIS access. Beyond that, topical worksheets that group first principles questions by difficulty tend to be far more useful for self-study than hunting through random past papers.