Key takeaways
- Calculus studies how things change (derivatives) and how they add up (integrals).
- A limit describes the value a function gets closer and closer to.
- The power rule: d/dx (xⁿ) = n·xⁿ⁻¹.
- Derivatives and integrals undo each other, like multiplication and division.
In this guide
What is calculus?
Calculus is the branch of maths that deals with change and accumulation. It was developed in the 1600s by Isaac Newton and Gottfried Leibniz, working separately.
It has two main parts. Differential calculus finds how fast something is changing at an instant. Integral calculus adds up many tiny pieces to find a total. Both are built on a third idea, the limit.
Why does calculus matter?
Algebra handles things that stay the same. Calculus handles things that change: a rocket speeding up, a population growing, medicine leaving the blood, water filling a tank.
Speed is the derivative of position. Distance is the integral of speed. That single pair of ideas powers physics, engineering and economics.
The three big ideas
| Idea | Question it answers | Notation | Example |
|---|---|---|---|
| Limit | What value does f(x) approach? | lim (x→c) f(x) | lim (x→1) (x² − 1)/(x − 1) = 2 |
| Derivative | How fast is it changing right now? | f′(x) or dy/dx | d/dx (x²) = 2x |
| Integral | How much has built up in total? | ∫ f(x) dx | ∫₀³ 2x dx = 9 |
The most useful rule to start with is the power rule for derivatives:
In words: bring the power down in front, then lower the power by one. And the Fundamental Theorem of Calculus links the two halves:
How to approach a calculus problem step by step
- 1Decide what is asked. A rate of change (derivative), a total or area (integral), or an approaching value (limit)?
- 2Write the function clearly, for example
s(t) = t²for position. - 3Pick the rule. Power rule, product rule, chain rule, or an integration method.
- 4Apply it carefully one term at a time.
- 5Substitute numbers if the question asks for a value at a point.
- 6Add units and interpret. Metres per second, square units, and so on.
You can check derivatives of powers with the power rule calculator and areas under power curves with the definite integral calculator.
A first look at limits
Try (x² − 1) ÷ (x − 1) at x = 1. You get 0 ÷ 0, which has no value. But factor the top: (x − 1)(x + 1) ÷ (x − 1) = x + 1. As x gets close to 1, this gets close to 2. That is the limit.
Common mistakes to avoid
- Forgetting + C. An indefinite integral always needs a constant:
∫ 2x dx = x² + C. - Mixing up derivative and integral. Derivative lowers the power, integral raises it.
- Plugging in too early. Find the derivative first, then put in the number.
- Treating 0 ÷ 0 as 0 or 1. It is undefined. Simplify first, then take the limit.
Tips and tricks
- Think in pictures. A derivative is the slope of a curve. A definite integral is the area under it.
- Units help. The derivative of metres with respect to seconds is metres per second.
- Explore limits with the limit calculator, and nested functions with the chain rule calculator.
Where you'll use it in real life
- Physics: speed and acceleration from position, as in the velocity calculator.
- Economics: marginal cost, the extra cost of making one more item.
- Biology and medicine: growth of populations and how fast a drug leaves the body.
- Engineering: finding the strongest shape or the lowest cost with the fewest materials.
Quick summary
Calculus studies change and accumulation. Limits describe what a function approaches, derivatives measure how fast it changes (start with d/dx (xⁿ) = n·xⁿ⁻¹), and integrals add up tiny pieces to find totals and areas. The two are opposites, linked by ∫ₐᵇ f(x) dx = F(b) − F(a). Use the free step-by-step calculators linked above to practise.
Frequently asked questions
What’s the difference between a derivative and an integral?
A derivative measures how fast something changes, like the slope of a curve. An integral adds things up, like the area under a curve. They are inverse operations.
When do I need calculus?
Calculus is used in physics, engineering, economics, biology and data science, whenever you study rates of change, totals built up over time, or the best (maximum or minimum) value.
Is calculus hard to learn?
The ideas are simple: slope and area. Most difficulty comes from weak algebra, so practise rearranging and simplifying expressions first.
What should I learn before calculus?
You need solid algebra, functions and graphs, and some basic trigonometry. Limits are usually the first calculus topic.
