Revise Limits and differentiation from first principles for IB Mathematics: Analysis and Approaches (5.1)

A limit describes how a function behaves as its input gets closer and closer to a particular value. Imagine walking along the graph of a function towards a specific $x$-coordinate: the limit is the $y$-value you are getting closer and closer to. Crucially, the limit does not care what the function's value is at the point, only what it approaches as you get near.

We write 'the limit of $f(x)$ as $x$ approaches $c$ is $L$' as $\lim_{x \to c} f(x) = L$. For most functions in this course the simplest way to find a limit is direct substitution: if putting $x=c$ into $f(x)$ gives a finite number, that number is the limit.

We already know how to find the gradient of a straight line: 'rise over run'. We can approximate the gradient of a curve by drawing a chord (a secant line) through two points on the curve, at $x$ and $x+h$. The 'run' is $(x+h) - x = h$ and the 'rise' is $f(x+h) - f(x)$. This gives the average rate of change between the two points.

Gradient of the chord: $m_{\text{chord}} = \frac{f(x+h) - f(x)}{h}$

Our goal is the gradient at a single point — the gradient of the tangent there. Imagine sliding the point at $x+h$ along the curve towards the point at $x$. As the two points close in, $h$ shrinks and the chord becomes a better and better approximation of the tangent. The tangent is the limiting position of the chord as $h \to 0$: the gradient of the curve at a point is the limit of the gradient of the chord.

Combining the chord gradient with the concept of a limit gives the formal definition of the derivative. The derivative of $f(x)$, written $f'(x)$ or $\frac{dy}{dx}$, is the gradient of the tangent at any point $x$ — a whole new function, the gradient function. Finding it from this definition is called 'differentiating from first principles'.

The derivative from first principles: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

The two notations mean the same thing. $f'(x)$ (Lagrange) is read 'f prime of x'; $\frac{dy}{dx}$ (Leibniz), for $y = f(x)$, is read 'the derivative of $y$ with respect to $x$' and is a reminder that the derivative is the limiting ratio of a small change in $y$ to a small change in $x$. Both give the gradient function, and both can be read physically as the instantaneous rate of change of $f$ with respect to $x$.

Once you have differentiated $x^2$ from first principles (giving $2x$) and $x^3$ (giving $3x^2$), a pattern emerges. Differentiating $x^n$ from first principles always produces the same shape of answer, and this is the power rule.

Power rule: $\frac{d}{dx}\left(x^n\right) = n x^{n-1}$, and $\frac{d}{dx}\left(a x^n\right) = a n x^{n-1}$ for a constant $a$.

So $\frac{d}{dx}(3x^2 - 2x) = 6x - 2$, matching the first-principles answer above. The power rule is a shortcut that saves you the limit every time — but it does not replace first principles. When a question demands 'from first principles', the power rule alone scores nothing; you must show the limit. A useful strategy is to work out the target answer with the power rule in your head first, so you know what you are aiming for as you carry out the limit.

In Paper 1 a 'from first principles' question is an answer-given style of task in spirit: the marks are analytic, tied to specific lines of working, and the credit lives in the method, not just the final expression. Each mark is either a method mark (M) — for the correct approach — or an accuracy mark (A) — for correct working or the correct answer, and an A mark is dependent on the M mark it follows. 'ISW' (ignore subsequent working) means a correct answer is not un-credited by later clutter, and 'FT' (follow-through) can carry an earlier slip forward. But because the answer is effectively given (you can get it instantly from the power rule), the working must be shown — an unsupported answer earns nothing. Study how each mark below is earned by a specific line.

First principles is the definition; the power rule is the shortcut it justifies. From here you will differentiate sums of powers, then products, quotients and composite functions with further rules — each one ultimately provable from the same limit. The two readings of the derivative you met here, gradient of a tangent and rate of change, drive everything that follows: finding equations of tangents and normals, locating maximum and minimum points where $f'(x)=0$, and modelling how quantities change in kinematics and optimisation. Master the habit — write the definition, substitute, expand, cancel, take the limit — and the calculus that follows becomes variations on a method you already own.