Calculus has a reputation, and the reputation does most of the damage. Students arrive expecting not to understand it, and when the first test goes badly they conclude they were right. The truth is more specific and more useful: the ideas of calculus are few and can be drawn, and what makes the course hard is the algebra inside every problem. This page says how hard each calculus course really is and what decides it for you.
The honest answer
Calculus is harder than any math course before it, for three reasons. The ideas are a new kind of thinking. The problems are longer. And it is the first course that uses everything you were taught earlier, all at once.
It is also more learnable than its reputation. There are only three ideas: the limit, the derivative and the integral. Each is a picture: a value being approached, a slope, an area. They are explained in what is basic calculus. A student who has those pictures and fluent algebra passes. A student with the pictures and shaky algebra understands every lesson and fails the tests.
Course by course
| Course | What it covers | How hard, for most students | Where the marks are lost |
|---|---|---|---|
| Calculus 1 | Limits, derivatives, applications (rates, maxima and minima), first integrals | The gentlest of the three | Algebra and trigonometry inside the problems |
| Calculus 2 | Techniques of integration, applications of integrals, sequences and infinite series | Usually the hardest | Choosing the technique; series tests; volume of memory work |
| Calculus 3 | Vectors, partial derivatives, multiple integrals, vector calculus | Often easier than Calculus 2 | Picturing three dimensions; setting up the region |
| Differential equations | Equations involving derivatives | Similar to Calculus 2 | Recognising the type |
Why Calculus 2 is the hard one. Differentiation has rules: follow them and you get the answer. Integration does not. Faced with an integral, you must decide which of several techniques to try, and nothing tells you which. Then the course turns to infinite series, which looks unlike anything before it. It is the course where practice counts most.
In high school. AP Calculus AB covers roughly Calculus 1. AP Calculus BC covers Calculus 1 and most of Calculus 2 in the same year, so it moves about half as fast again. BC is harder because of pace and the series unit, not because its students are different people. In the UK, A-level maths spreads Calculus 1 content across two years, with further maths going beyond it. In CBSE, Class 11 introduces limits and derivatives and Class 12 does most of Calculus 1 and part of Calculus 2.
Four things that decide how hard it is for you
- How automatic your algebra is. Factoring, exponent rules including negative and fractional powers, fractions with letters, solving equations. This is by far the largest factor.
- Trigonometry. The unit circle, the basic identities, radians. Calculus uses them constantly from the first month.
- Whether you practise daily. Calculus is a skill, like a language or an instrument. Twenty minutes a day beats three hours on Sunday.
- The pace. A college semester or a BC year leaves no room to fall a week behind. A summer course at double speed is harder still.
Talent comes a long way down this list.
Calculus against other courses
| Compared with | Which is harder? |
|---|---|
| Algebra 2 | Calculus, for most; though some find Algebra 2 harder because it is a collection of unrelated topics |
| Precalculus | Similar workload. Precalculus has more to memorise; calculus has more reasoning. Students who did well in precalculus are usually fine |
| Trigonometry | Calculus. But weak trigonometry is one of the main reasons calculus feels hard |
| Statistics | Calculus needs more algebra; statistics needs more judgement about meaning. Different, not ranked |
| Physics | They help each other. Physics gives calculus something to be about, and calculus makes physics simpler |
| Linear algebra | More abstract than calculus, with less computation |
The comparison with algebra, with one problem solved both ways, is in the difference between algebra and calculus.
A readiness check
No calculator, ten minutes.
- Factor x² − 7x + 12
- Simplify (x² − 9) / (x + 3)
- Write √x and 1/x² as powers of x
- Solve 2x² − 5x − 3 = 0
- Expand (a + b)³
- What are sin 0, sin 90° and cos 180°?
- Convert 60° to radians
- If f(x) = 3x² − 2, find f(x + h)
- Solve for x: 2ˣ = 32
- Sketch y = x² − 4 and mark where it crosses the x-axis
Answers: 1) (x − 3)(x − 4) 2) x − 3 3) x to the power ½, and x⁻² 4) x = 3 or x = −½ 5) a³ + 3a²b + 3ab² + b³ 6) 0, 1 and −1 7) π/3 8) 3x² + 6xh + 3h² − 2 9) x = 5 10) a U-shaped curve crossing at −2 and 2, lowest point at (0, −4).
Nine or ten, quickly: calculus will be work, not a wall. Six to eight: spend a month on the ones you missed before the course starts. Five or fewer: take or retake precalculus first; it will save you a failed course.
The month before you start
- Week 1: exponents and factoring, twenty questions a day.
- Week 2: fractions with letters, and solving quadratics three ways.
- Week 3: the unit circle and the six basic trigonometric values, from memory.
- Week 4: functions: f(x + h), composition, sketching the basic curves.
Then, once the course begins: draw every problem, write every step, check every answer, and do some every day. How long the whole subject takes is in how long does it take to learn calculus.
When it is going badly
If you are already in the course and failing tests you thought you understood, do not re-read the chapter. Take one marked test, and for each lost mark write down whether the error was calculus or algebra. Nearly always, most of them are algebra. That list is what to practise.
Where to get help
A one-to-one online calculus class with a tutor who holds a master's in mathematics: the readiness check in the first class, the algebra and trigonometry gaps closed alongside the course, and every line of your working read. For AP Calculus AB and BC, Calculus 1 to 3 at college, A-level and CBSE. The free 30-minute first class is a real lesson on the problem you bring.
Questions parents ask
1Is calculus hard?
2How hard is Calculus 1?
3Is Calculus 2 harder than Calculus 1?
4Is Calculus 3 harder than Calculus 2?
5Is calculus harder than statistics?
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