Calculus is the branch of mathematics that studies how quantities change. Its two main tools are the derivative, which measures the rate of change at an instant (the slope of a curve at a point), and the integral, which adds up infinitely many tiny pieces (the area under a curve). Both are built on limits.
In Spanish
cálculo
Also called
differential calculus, integral calculus, calc, the mathematics of change
Grade level
Calculus
CalculusAP
Calculus in a picture
Calculus asks two questions about a curve: how steep is it at one point (the derivative) and how much area lies under it (the integral). Limits answer both.
Formal definition and notation
Calculus is the study of functions through limits. Differential calculus defines the derivative f′(a) = limh → 0f(a + h) − f(a)h, the instantaneous rate of change. Integral calculus defines the definite integral as the limit of sums of thin rectangles, which gives the area under a curve. The Fundamental Theorem of Calculus says these two operations undo each other.
Symbol
Read it as
Meaning
limx → af(x)
the limit of f of x as x approaches a
the value f(x) gets close to as x gets close to a
f′(x) or dydx
f prime of x, or d y d x
the derivative: rate of change, slope of the tangent line
∫ f(x) dx
the integral of f of x d x
the integral: accumulated total, area under the curve
Key facts
Two big questions: how fast is something changing right now (derivatives), and how much has built up in total (integrals)?
Algebra finds the slope of a straight line; calculus finds the slope of a curve at a single point.
Speed is a derivative: if position is s(t) = t2meters, the speed at t = 3 seconds is s′(3) = 6 meters per second.
Area under a speed graph is an integral: it gives the distance traveled.
In US schools, calculus usually comes after precalculus.
Where the name comes from. The modern tools of calculus came from the independent work of Isaac Newton and Gottfried Leibniz; mathematicians such as Weierstrass later put it on a rigorous footing.
Calculus examples, step by step
Example 1: Average rate of change (the algebra step)
Answer: The ball is rising at 10 meters per second at t = 1.
Example 3: Area under a line with an integral
Test-style
A car’s speed is v(t) = 3t meters per second for 0 ≤ t ≤ 10. How far does it travel in those 10 seconds?
Distance is the area under the speed graph from t = 0 to t = 10.
The region is a triangle with base 10 and height v(10) = 30.
Area = 12 × 10 × 30 = 150.
An integral gives the same: the antiderivative of 3t is 1.5t2, and 1.5 × 100 = 150.
Answer: The car travels 150 meters.
Common mistakes
✗ MistakeCalculus is just harder algebra.
✓ InsteadCalculus adds a new idea, the limit, which algebra does not have.
Algebra can find the slope between two points. Calculus lets the two points move together until they become one point, which needs limits.
✗ MistakeThe speed at t = 3 for s(t) = t2 is s(3)3 = 3.
✓ InsteadThe speed is s′(3) = 2 × 3 = 6.
s(3)3 is the average speed from 0 to 3. The speed at the instant t = 3 is the derivative.
✗ MistakeThe derivative of x3 is 3x3.
✓ InsteadThe derivative of x3 is 3x2.
The power rule brings the exponent down as a factor and lowers the exponent by 1.
Not to be confused with
Calculusvs.Precalculus
Precalculus reviews functions, trigonometry and graphs to prepare for calculus; it does not use limits, derivatives or integrals as its main tools.
Calculusvs.Calculator
A calculator is a device for computing. Calculus is a branch of mathematics; a calculator can evaluate a derivative, but calculus is the set of ideas behind it.
The derivative is one tool inside calculus. Calculus also includes limits, integrals and series.
Quick check: 3 questions
Question 1 of 3
What does a derivative measure?
Show answer and explanation
Answer: B. A derivative is the rate of change at one instant, the slope of the tangent line. The area under a curve is an integral; the other two are not calculus ideas.
Question 2 of 3
For f(x) = x2, what is the slope of the tangent line at x = 4?
Show answer and explanation
Answer: D.f′(x) = 2x, so f′(4) = 8. 16 is f(4), the height of the curve, not its slope.
Question 3 of 3
What is the area under y = 2 from x = 0 to x = 5?
Show answer and explanation
Answer: A. The region is a rectangle 5 wide and 2 tall, so the area (the integral of 2 from 0 to 5) is 10. 7 adds the sides instead of multiplying.
Calculus: frequently asked questions
What is calculus used for in real life?
Anywhere something changes. Physics uses derivatives for velocity and acceleration and integrals for work and distance. Engineers use calculus to design curves and structures, economists use it for marginal cost and profit, and doctors use it in models of how a medicine’s concentration changes in the body over time.
Is calculus hard?
Calculus combines skills from algebra, functions, graphing and trigonometry, so gaps in those topics make it feel hard. The new idea itself, the limit, is very intuitive: zooming in until a curve looks straight. Students who are fluent with slope, function notation and algebraic simplifying usually find the main rules, like the power rule, quick to learn.
What do you need to know before calculus?
Strong algebra (factoring, exponents, solving equations), functions and their graphs, slope and rate of change, and trigonometry with the unit circle. Most US students take precalculus first. Understanding slope as rise over run is especially important, because a derivative is the slope of a curve at a single point.
DerivativeA derivative measures how fast a function’s output changes at one exact input: its instantaneous rate of change. On a graph it is the slope of the tangent line at that point. The derivative of f(x) = x2 is f′(x) = 2x, so at x = 3 the curve is rising with slope 6.Calculus · CalculusLimitA limit is the value a function gets closer and closer to as the input approaches a certain number. We write limx → af(x) = L. The function need not even be defined at a. For f(x) = x2 − 1x − 1, f(1) is undefined, but the limit as x → 1 is 2.Calculus · Precalculus, CalculusRate of ChangeA rate of change tells how much an output changes for each 1-unit change in the input: the change in y divided by the change in x. If a phone plan costs 15 dollars more for every extra hour, its rate of change is 15 dollars per hour. For a straight line, the rate of change equals the slope.Functions · Grade 8, Algebra 1, Algebra 2IntegralAn integral adds up infinitely many tiny pieces to find a total, such as the area under a curve. The definite integral ∫baf(x) dx is the signed area between the graph of f and the x-axis from x = a to x = b. An indefinite integral is an antiderivative: ∫ 2xdx = x2 + C.Calculus · CalculusSlopeSlope is a number that measures how steep a line is: the change in y divided by the change in x between any two points on the line, often called rise over run. A positive slope rises from left to right, a negative slope falls, and a horizontal line has slope 0.Algebra · Grade 8, Algebra 1MeterA meter (m) is a metric unit of length equal to 100 centimeters. It is the base unit of length in the metric system, a little over three feet (about 3.28 feet). A doorway is about 2 meters tall, and 1,000 meters make a kilometer. To change meters to centimeters, multiply by 100.Measurement · Grade 3, Grade 4, Grade 5FunctionA function is a rule that gives exactly one output for each input. If you put in one x-value, a function gives back one y-value. In f(x) = 2x + 1, the input 3 gives the output 7. Two inputs may share an output, but one input can never have two different outputs.Functions · Grade 8, Algebra 1, Algebra 2TrigonometryTrigonometry is the branch of math that studies how the angles and the side lengths of triangles are related. Its three main ratios, sine, cosine and tangent, compare two sides of a right triangle. Knowing one side and one acute angle, or two sides, you can find every other side and angle without measuring them.Trigonometry · Geometry, Algebra 2, PrecalculusAverage Rate of ChangeThe average rate of change of a function over an interval is the change in output divided by the change in input: f(b) − f(a)b − a. It tells how fast the function changes, on average, from x = a to x = b. On a graph it is the slope of the line through the two endpoints.Functions · Algebra 1, Algebra 2, PrecalculusAverageAn average is one number that stands for the typical size of the values in a data set. In everyday math, average means the mean: add all the values, then divide by how many values there are. In statistics, the median and the mode are also kinds of average.Statistics · Grade 5, Grade 6, Grade 7SecantSecant (sec) is the trigonometric function equal to 1 divided by cosine: sec θ = 1cos θ. In a right triangle it is the hypotenuse divided by the side adjacent to the angle. Secant is undefined wherever cosine is 0, such as at 90°, and its values are always at least 1 or at most −1.Trigonometry · Algebra 2, Precalculus, CalculusHeightHeight is how tall something is: the distance from its bottom to its top, measured straight up and down. In geometry, the height of a shape is the perpendicular distance from a base to the opposite vertex or side, so it always meets the base at a right angle. Area and volume formulas use it.Measurement · Grade 2, Grade 5, Grade 6Power RuleThe power rule is the shortcut for the derivative of a power of x: if f(x) = xn, then f′(x) = nxn − 1. Bring the exponent down as a coefficient and subtract 1 from it, so x5 becomes 5x4. It works for any real exponent, including negative and fractional ones.Calculus · CalculusSubstituteSubstitute means to replace a variable with a number or with an equal expression. In 4x − 1, substituting 3 for x gives 4(3) − 1. You substitute to evaluate expressions, to check whether a number is a solution, and to solve systems of equations by swapping in an equal expression.Algebra · Grade 6, Grade 7, Grade 8TriangleA triangle is a polygon with three sides and three angles. Its three corners are called vertices, and the angle measures in any triangle add to 180 degrees. Triangles are classified by their sides (equilateral, isosceles, scalene) and by their angles (acute, right, obtuse).Geometry · Grade 3, Grade 4, Grade 5AreaArea is the amount of surface a flat shape covers, measured in square units. It counts how many unit squares fit inside the shape, which is why the units are squared. A rectangle’s area is length times width, so a 5 by 3 rectangle has area 15 square units.Geometry · Grade 3, Grade 4, Grade 5GraphA graph is a visual display of numbers, relationships, or data. Graphs help you see patterns, compare values, and understand relationships quickly. In algebra, the graph of an equation is the picture of all its solutions on a coordinate plane; in statistics, bar graphs, line graphs and scatter plots display data.Functions · Grade 5, Grade 6, Grade 7