IB Maths AI HLDifferentiationPaper 1 & 2~8 min read
Introduction to Derivatives
Calculus is all about rates of change. The derivative of a function is a new function that hands you the gradient at every value of x. It’s built from a limit: slide a chord’s far end towards a fixed point and its gradient settles on the gradient of the tangent โ and that limiting gradient is exactly what the derivative gives you.
๐ What you need to know
Limit: the value a function approaches as x nears a particular value from either side โ useful when the function is undefined there.
Derivative: a function that outputs the gradient of y = f(x) at each x. Also called the gradient function.
Gradient at a point = gradient of the tangent at that point.
Chord โ tangent: the gradient of a chord [PQ] tends to the gradient of the tangent at P as Q slides towards P.
Notation: for y = f(x), the derivative is dydx = fโฒ(x). Other variables work the same way.
GDC: estimate limits from tables/graphs; chord gradients sneak up on the tangent gradient.
What is a limit?
The limit of a function is the value it heads towards as x approaches some number from both sides. Limits matter most exactly where the function can’t be evaluated directly โ typically a “0/0” situation.
Classic examplef(x) = x4 โ 1x โ 1 โ 4 as x โ 1
Undefined at x = 1 (รท0) โ but the limit still exists โ
At x = 1 the formula tries to divide by zero, so f(1) is undefined. Yet plug in values creeping towards 1 from below (0.9, 0.99, โฆ) and from above (1.1, 1.01, โฆ) and the outputs close in on 4 from both sides. That target value is the limit.
๐ค Why bother with limits?
A gradient is “rise over run” between two points. To get the gradient at a single point you’d need the two points to be the same โ but that gives 0/0, which is meaningless on its own. Limits are the tool that lets you ask “what does the chord gradient approach as the two points merge?” without ever actually dividing by zero. Every derivative is secretly a limit.
In the exam: you may be asked to predict or estimate a limit from a table of function values or from a graph on your GDC โ not to prove it algebraically. Read off the value both sides are converging on.
What is a derivative?
Calculus is the study of how things change. The way a car’s position changes is its speed; the way its speed changes is its acceleration. For a curve, the thing that changes is the gradient โ and unlike a straight line, the gradient of a curve is different at every x.
the functionf(x)Input an x, get a height (y-value) on the curve.
the derivativefโฒ(x)Input an x, get the gradient (steepness) of the curve there.
So the derivative is itself a function: feed it a value of x and it returns the gradient of the original curve at that point. That’s why it’s also called the gradient function.
How limits and derivatives link up
Fix a point P on the curve y = f(x). Draw chords from P to nearby points Q1, Q2, โฆ further along the curve. As Qslides down towards P, each chord tilts closer to the tangent, and the chord gradients home in on the gradient of the tangent at P.
Chords closing in on the tangent at P
As Q slides down to P, the red chords rotate onto the purple tangent โ their gradients tend to the tangent’s gradient.
๐งญ Recipe โ estimate a gradient at a point
Fix P at the x-value you care about; note its coordinates.
Pick points Q on the curve getting progressively closer to P.
Find each chord gradient using y2 โ y1x2 โ x1.
Spot the limit: the value the gradients are settling towards is your estimate of the tangent gradient at P.
๐ง “Derivative = Direction”
Both start with D. The derivative tells you which direction (and how steeply) the curve is heading at each point โ positive means uphill, negative means downhill.
Notation for derivatives
For y = f(x), the derivative with respect to x is written two equivalent ways. Whatever letters a question uses, the bottom of the d-fraction is the variable you’re differentiating with respect to.
Two ways to write the same thingdydx = fโฒ(x) | if V = f(s) then dVds = fโฒ(s)
Notation expected throughout the course โ
Worked examples
All five use the SME curve y = f(x) where f(x) = x3 โ 2, passing through P(2, 6), A(2.3, 10.167), B(2.1, 7.261) and C(2.05, 6.615125).
WE 1
Find the gradient of chord [PA]
Use the gradient-of-a-line formula with P(2, 6) and A(2.3, 10.167).
Use the trend in the chord gradients as the x-gap shrinks to zero.
chords: 13.89 โ 12.61 โ 12.3025 โฆgradients fall and level off near 12 as ฮx โ 0tangent gradient at x = 2 โ 12there’s a limit the chord gradient reaches as the x-difference approaches zero.
WE 5
Predict the limit of f(x) = (x4 โ 1)/(x โ 1) as x โ 1
The function is undefined at x = 1 โ estimate the limit from values either side.
x = 0.99 โ 3.9404 โฆ x = 1.01 โ 4.0604 โฆboth sides squeeze towards 4 as x โ 1limit = 4f(1) itself is undefined (รท0), but the limit exists.
๐ก Top tips
Gradient at a point = gradient of the tangent there โ keep that link front of mind.
Closer Q, better estimate: the smaller the x-gap, the nearer the chord gradient is to the true tangent gradient.
Approach from both sides when estimating a limit โ the two sides should agree.
Undefined โ no limit: a function can be undefined at a point yet still have a clear limit there.
Match the letters: dVds means differentiate V with respect to s โ don’t default to x.
Use your GDC to tabulate or graph values when asked to estimate a limit.
โ Common mistakes
Flipping the gradient fraction โ it’s (change in y) over (change in x), not the other way round.
Subtracting in inconsistent order โ keep P first (or Q first) on top and bottom.
Claiming the gradient equals a chord โ a chord only estimates the tangent; the true gradient is the limit.
Saying the limit “doesn’t exist” because f is undefined โ the limit is about the approach, not the value at the point.
Confusing f(x) with fโฒ(x) โ one gives heights, the other gives gradients.
Next up โ Differentiating Powers of x. Estimating gradients chord-by-chord is fine for building intuition, but it’s slow. The next topic gives you the shortcut: a single rule (f(x) = xn โ fโฒ(x) = nxnโ1) that turns the whole limiting process into one line of algebra, so you can write down fโฒ(x) instantly and then plug in any x to get the gradient there.
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