Calculus is one of the most learnable subjects to teach yourself, because there are only three ideas in it and each can be drawn. It is also one of the most abandoned, because most people who try start in the wrong place, practise the wrong way, and conclude after a month that calculus is beyond them when what defeated them was algebra. This is the order that works, with a plan.
Before you start: the one check that matters
Nearly every self-taught student who gives up on calculus gives up on the algebra inside it. Ten minutes, no calculator:
- Factor x² − 5x + 6
- Simplify (x² − 4) / (x − 2)
- Write 1/√x as a power of x
- Solve 3x² − x − 2 = 0
- If f(x) = x² + 1, write out f(x + h) − f(x) and simplify
- What are sin 30°, cos 60° and tan 45°?
- Convert 45° to radians
- Sketch y = 1/x and y = eˣ, roughly, on the same axes
Answers: 1) (x − 2)(x − 3) 2) x + 2 3) x to the power −½ 4) x = 1 or x = −⅔ 5) 2xh + h² 6) ½, ½ and 1 7) π/4 8) a curve in two pieces that never touches the axes; a curve that rises steeply to the right and flattens toward zero to the left.
Seven or eight right, quickly: start calculus now. Five or six: spend two weeks on the ones you missed first. Four or fewer: a month on algebra and trigonometry before calculus will save you three months of frustration inside it. What calculus assumes, and why, is in is calculus hard.
The order to learn the ideas in
Calculus 1 has three ideas, and they must be learned in this order because each uses the one before.
1. Limits (one week). The idea of a value being approached rather than reached. Learn it as a picture: a point on a curve that you walk toward. Then the handful of rules for evaluating limits, including the trick of cancelling a factor that makes a fraction 0/0. Do not spend long on the formal definition; you can return to it later. The limit is here because the derivative is defined with one.
2. Derivatives (four to five weeks). The slope of a curve at a point. Learn it first as the slope of a line through two points getting closer together, do the definition by hand for x² and x³ so that you have seen where the rules come from, then learn the rules: power, product, quotient, chain. The chain rule is the one that needs the most practice. Then the uses: where a function is increasing or decreasing, maxima and minima, related rates, and sketching curves. Half of Calculus 1 is here.
3. Integrals (four to five weeks). The area under a curve, first as the sum of thin rectangles, then as the reverse of differentiation through the fundamental theorem, which is the most important single result in the course. Then substitution, which is the chain rule run backwards, then the uses: areas between curves, volumes, average value.
Each idea in words and with a worked example is in what is basic calculus. Read that first if the three words above mean nothing to you yet.
How to use a textbook, and how to use videos
The textbook is where you learn. Choose one with answers to the odd-numbered exercises and a worked example before each exercise set; any standard college calculus book or an AP Calculus textbook will do. Read a section, then do the exercises: not three, but twenty, until the method is automatic and not just understood. Check against the answers. For each wrong answer, find the step that went wrong before moving on.
Videos are where you get introduced. A good video lesson makes a new idea clear in twenty minutes, which a textbook sometimes cannot. But watching a video does not teach you to do anything, and the feeling of understanding it gives is the single biggest trap in self-study. Rule: for every twenty minutes of video, forty minutes of problems from the book on the same topic, that day.
Worked examples are to be attempted, not read. Cover the solution. Try it. Compare. If you could not start, read the first line only, cover it, and continue.
A twelve-week plan for Calculus 1
An hour a day, six days a week. This covers Calculus 1 and most of AP Calculus AB for a student who passed the check above.
| Week | Topic | What you should be able to do by the end of the week |
|---|---|---|
| 1 | Limits | Evaluate limits by substitution, by cancelling and from a graph; recognise where a limit does not exist; limits at infinity |
| 2 | The derivative as a limit; the power rule | Find a derivative from the definition for a polynomial; use the power rule fluently, including negative and fractional powers |
| 3 | Product, quotient and chain rules | Differentiate any combination of polynomials, roots and fractions; the chain rule without hesitation |
| 4 | Trigonometric, exponential and logarithmic derivatives | The six basic derivatives from memory; combined with the chain rule |
| 5 | Implicit differentiation and related rates | Differentiate a curve given as an equation in x and y; set up and solve a related-rates problem from a diagram |
| 6 | Increasing and decreasing, maxima and minima | Find and classify stationary points; solve an optimisation problem from a word description |
| 7 | Curve sketching; review | Sketch a function from its derivatives; a full timed review of weeks 1 to 6 |
| 8 | The integral as area; antiderivatives | Reverse the power rule; estimate an area with rectangles; the notation |
| 9 | The fundamental theorem; definite integrals | Evaluate a definite integral; understand why the area under a rate gives a total |
| 10 | Substitution | Recognise when substitution applies and carry it through, including changing the limits |
| 11 | Applications: area between curves, volumes, average value | Set up the integral from a sketch, which is where the marks are |
| 12 | Full review; timed practice | Past-paper or released exam questions under time; the error log reread |
Keep an error log from day one: each wrong problem in one line, with whether the error was calculus or algebra. By week four it will tell you what to practise separately, and for most students the answer is algebra.
How to learn calculus fast
Sometimes there is no twelve weeks: a course starts in a month, or an exam is six weeks away. Here is what to cut and what to keep.
- Keep: the three ideas as pictures; the derivative rules; the uses of the derivative (maxima and minima, related rates); the fundamental theorem; substitution.
- Cut, for now: the formal definition of a limit; proofs of the rules; unusual limits; the longer integration techniques (parts, partial fractions, trigonometric substitution belong to Calculus 2 anyway); the more exotic applications.
- Pace: limits in three days, derivatives and their rules in ten days, uses of the derivative in a week, integrals and the fundamental theorem in a week, substitution and applications in a week. That is four to five weeks at ninety minutes a day.
- The condition: fluent algebra. Learning fast is impossible while also learning algebra. If the check at the top of this page went badly, the fast route is closed; take the slower one.
Fast learning gives you the ideas and the standard methods. It does not give you exam fluency, which comes only from timed practice on real papers, so leave the last week for that. How long each course takes, case by case, is in how long does it take to learn calculus.
If you are learning it for a particular exam
| Exam | What to add to the plan above |
|---|---|
| AP Calculus AB | Released free-response questions from week 7 onward, scored against the College Board rubric; the calculator-allowed and no-calculator sections practised separately |
| AP Calculus BC | Everything in AB, then series (the hardest unit and the real difference), parametric and polar functions, integration by parts and partial fractions; plan on an extra six weeks |
| A-level maths | The pure papers; differentiation and integration are spread over two years, so the plan above goes further than the first year needs; add the exam board's own past papers |
| CBSE Class 12 | NCERT chapters 5 to 8 in order; the NCERT exercises and the board's sample papers, because the questions come from there |
| College placement or a Calculus 1 final | The plan as it stands, with the last two weeks on past finals from your college if they are published |
The five mistakes that waste months
- Starting calculus with shaky algebra. The single biggest cause of giving up. The check at the top is there for a reason.
- Watching instead of doing. Forty hours of video lessons and no problems produces a student who recognises every idea and can solve nothing.
- Learning rules without pictures. The derivative is a slope and the integral is an area. A student who has the rules but not the pictures cannot set up a word problem, which is most of the exam.
- Rushing the chain rule. It appears in nearly every later problem, in differentiation and, backwards, in substitution. Two extra days on it in week three save weeks later.
- Studying once a week. Calculus is a skill. Six hours on Sunday are remembered worse than one hour on six days; the spacing effect is one of the most reliable findings in the psychology of learning.
Where to get help
Self-study stalls at the same place for everyone: a problem you cannot do and cannot see why. A 1-on-1 online calculus class once a week with a tutor who holds a master's in mathematics unsticks it: your week's error log read line by line, the algebra gaps fixed alongside the calculus, and the next week planned. For AP Calculus AB and BC, Calculus 1 to 3, A-level and CBSE Class 12, and for adults learning it for a degree or a career. The free 30-minute first class is a real lesson on the problem you bring.
Questions parents ask
1How do I learn calculus on my own?
2Can I learn calculus by myself?
3How long does it take to learn calculus?
4How can I learn calculus fast?
5What do I need to know before learning calculus?
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