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Everything you need to actually understand calculus — from limits to multivariable — on one page. Live graphs you can drag, endless practice problems with step-by-step solutions, and real-world examples that make it click.
The mathematics of change and accumulation.
Algebra and geometry describe a world that holds still — a line has one slope, a rectangle has one area. But the real world moves: cars accelerate, populations grow, temperatures swing, prices fluctuate. Calculus is the math of things that change.It gives you two superpowers, and one beautiful theorem that says they're secretly the same power.
How fast is something changing right now? The derivative measures an instantaneous rate of change — the slope of a curve at a single point. Speed from position, growth rate from population, marginal cost from total cost.
How much has accumulated in total? The integral adds up infinitely many tiny pieces — the area under a curve. Distance from speed, total revenue from sales rate, volume from cross-sections.
The clever move behind both ideas is the limit: instead of computing a slope over a gap or an area with chunky rectangles, we ask what happens as the gap shrinks to zero and the rectangles become infinitely thin. The limit turns an approximation into an exact answer. That's why we start with limits in Chapter 1.
Work top to bottom, or jump around with the contents menu. Every blue-ish box is interactive — drag the sliders and points. Every you see is real, computed live. Hit “New problem” in any practice box for unlimited randomized exercises with full solutions. Mark sections done to track your progress.
You don't need to be a wizard — but these five things show up constantly.
Most “calculus is hard” moments are really “my algebra got rusty” moments. Skim these. If a box looks obvious, great — close it and move on. If not, this is the stuff worth shoring up first.
A function is a machine: feed it an input, get one output. means “square the input, add 1.” So . We'll constantly plug things in — including expressions like , which just means “replace every x with (x+h).”
You can start calculus today. Refer back here whenever an algebra step trips you up — that's more efficient than trying to “finish” reviewing first.
The idea that makes everything else in calculus possible.
A limitanswers the question: “as the input gets arbitrarily close to some value, what does the output head toward?” Crucially, we don't care what happens at the point — only what happens near it. That distinction is the whole game.
Read aloud: “the limit of as approaches equals .”
| x | f(x) |
|---|---|
| 0.5 | 1.5 |
| 0.8 | 1.8 |
| 0.95 | 1.95 |
| 0.99 | 1.99 |
| x | f(x) |
|---|---|
| 1.5 | 2.5 |
| 1.2 | 2.2 |
| 1.05 | 2.05 |
| 1.01 | 2.01 |
The function is undefined at x = 1 (the hole), yet both sides head to the same height. That shared height is the limit: .
Sometimes the left and right approaches disagree. We write for the approach from the left and from the right. The two-sided limit exists only when both sides agree:
A function is continuous at if you can draw through it without lifting your pen — formally, . Here are the classic ways that fails:
Limit exists, but f(a) is missing or misplaced.
Left and right limits disagree.
Function blows up to ±∞ (a vertical asymptote).
Direct substitution into gives — indeterminate. Factor and cancel:
is an indeterminate form— it's a signal that you have more work to do (factor, rationalize, or use L'Hôpital's rule later), not an answer. Same goes for and .
For end behavior of rational functions, compare the degrees of the top and bottom. A quick rule:
Evaluate the limit.
Hold onto the idea of "a gap shrinking to zero." In the next chapter we apply a limit to the slope formula and out pops the derivative.
Instantaneous rate of change — the slope of a curve at a single point.
Take the slope formula and shrink the run to zero. The slope between two points (a secant) becomes the slope at one point (a tangent). That limit is the derivative:
Drag the gap toward zero below and watch the red secant snap onto the teal tangent:
As h → 0, the red secant slope approaches the teal tangent slope — that limit is the derivative.
Let's find for using only the limit — no shortcuts. It's worth doing by hand once, because it shows why the power rule works.
Expand , and the terms cancel:
Factor out and cancel it (legal, since means ):
The power rule gives the same answer instantly: . Every shortcut rule is really just this limit, pre-computed.
Leibniz's is handy because it names the variables; Lagrange's is compact. Use whichever the problem makes natural.
The derivative is itself a function — it reports the slope at every . Drag across the graph: where the curve climbs, is positive; at peaks and valleys, ; where it falls, is negative.
The last two numbers are always equal — that's the definition of the derivative, made visible.
If is an object's position, then is its velocity and is its acceleration. Your car's speedometer is literally computing a derivative of your odometer reading in real time.
You almost never compute the limit by hand. Instead you learn rules. The workhorse:
“Bring the exponent down front, then subtract one from it.” Combined with the fact that constants pull out and sums differentiate term-by-term, you can already handle any polynomial.
Each term: , , and the constant vanishes (a flat line has zero slope).
Differentiate using the power rule.
Products, quotients, compositions, trig, exponentials, logs — the full toolkit is in Differentiation Rules.
The complete toolkit — combine these to differentiate anything.
Three structural rules handle how functions are combined, plus a short table of known derivatives. Master these and you can differentiate essentially any expression you'll meet.
“First times derivative of second, plus second times derivative of first.”
“Low d-high minus high d-low, over low squared.”
“Derivative of the outside (leave inside), times derivative of the inside.”
Forgetting to multiply by the inside's derivative is the most common calculus mistake. is , not . Whenever a function is “wrapped” around something other than a bare , the chain rule is in play.
| f(x) | f ′(x) |
|---|---|
Differentiate (product rule):
Differentiate (chain rule, inside is ):
Differentiate using the product rule.
Differentiate using the quotient rule. Simplify.
Differentiate using the chain rule.
Differentiate.
Differentiate again to get (acceleration, concavity), and again for , and so on. Each derivative describes how the previous one is changing.
Where derivatives earn their keep: shape, motion, and optimization.
Once you can find and , you can describe the entire shape of a graph, locate maxima and minima, relate changing quantities, and approximate hard values. This is the most useful chapter in all of differential calculus.
f ′ > 0 → curve rising; f ′ < 0 → curve falling. Where the curve levels off — a candidate max or min (a critical point).
f ″ > 0 → concave up (cup ∪); f ″ < 0 → concave down (cap ∩). Where concavity flips is an inflection point.
Optimization is the real-world payoff: maximize volume, minimize cost, find the best angle. The recipe is always the same — write the quantity as a function of one variable, differentiate, set , and check it's really a max or min.
From a 20×16 cm sheet, cut a square of side x from each corner and fold up the sides. What x gives the biggest box?
The maximum sits at x ≈ 2.94, exactly where the volume curve flattens — i.e. where V′(x) = 0. That's how optimization works: set the derivative to zero.
When two quantities are linked, their rates of change are linked too. Differentiate the relationship with respect to time and solve for the rate you want.
A circular slick's radius grows at m/s. How fast is the area growing when ?
The chain rule is doing the heavy lifting: depends on , which depends on .
Find the equation of the tangent line (solve for y).
Find the x-coordinate of the critical point (where f′(x) = 0).
A circle's radius grows at dr/dt = 4. Find dA/dt when r = 7, as a multiple of π — enter just the coefficient.
Find the second derivative.
In economics, the derivative of a total is called the marginal quantity: marginal cost is , marginal revenue is . Profit is maximized where marginal revenue equals marginal cost — a derivative set to zero.
Adding up infinitely many slivers — and the theorem that ties it all together.
Differentiation breaks change into instantaneous rates. Integration runs the other direction: it accumulates. The definite integral measures the area under a curve — and area turns out to encode totals of every kind: distance from speed, work from force, profit from marginal profit.
Approximate the area under a curve with rectangles, then let their number go to infinity. The limit of these Riemann sums is the definite integral:
Push nhigher and the rectangles squeeze the error toward 0. The exact area is the limit of these sums — that's the definite integral.
Here is the punchline of the entire subject — the bridge between derivatives and integrals. It comes in two parts:
The rate at which area accumulates is just the height of the curve.
where is any antiderivative of . Find an antiderivative, plug in the bounds, subtract.
Computing area by hand means summing infinitely many rectangles — brutal. Part 2 says you can skip all of that: just reverse-engineer a function whose derivative is , and evaluate it at the two endpoints. Two centuries of hard area problems collapse into "find an antiderivative."
An antiderivative (or indefinite integral) of is a function whose derivative is . Because the derivative of a constant is zero, antiderivatives always carry a :
On an indefinite integral, dropping the is an automatic deduction — there are infinitely many antiderivatives, differing by a constant. (On a definiteintegral the constant cancels in the subtraction, so you don't need it there.)
Find the indefinite integral (don't forget + C).
Evaluate the definite integral.
Antiderivatives don't follow neat rules like derivatives — here's the toolkit.
Differentiation is mechanical; integration is more like puzzle-solving. There's no universal product or quotient rule for integrals. Instead you learn a handful of techniques and recognize which one fits.
When you spot a function and its derivative inside the integral, substitute. It undoes the chain rule.
For , let , so :
Integrate using u-substitution (+ C).
For products of unlike functions (a polynomial times a log or exponential, say):
Choose by priority — Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential. Whatever's earlier in LIATE becomes ; the rest is .
: let (algebraic), . Then , :
| Integral | Result |
|---|---|
A few innocent-looking integrals — like (the bell curve!) — simply can't be written with elementary functions. That's not a failing on your part; it's a fact of life, and why numerical integration (like the Riemann sums earlier) matters.
Areas, volumes, averages, and totals — integrals everywhere.
If a derivative is a rate, an integral is a total. Anything built by accumulating a rate over an interval is an integral in disguise.
Integrate the gap across the interval where one curve sits above the other.
Set the bounds to the intersection points (a ≈ -3.37, b ≈ 2.37) to capture the whole enclosed region.
Spin a region around an axis and you get a solid. Slice it into thin disks of radius and thickness ; each disk has volume . Add them up:
Revolving on around the x-axis sweeps out a paraboloid. The dashed mirror curve shows the bottom of the solid.
The shell method (cylindrical shells ) is the alternative when revolving around a vertical axis.
It's the height of the rectangle with the same area as the region under the curve — the continuous version of an average.
Evaluate the definite integral.
Total distance traveled is the integral of speed: . Work done by a variable force is . Charge is the integral of current; volume of water is the integral of flow rate. The same operation, dressed in different physics.
Even the length of a curve is an integral: . Notice it uses the derivative inside an integral — the two halves of calculus working together.
Adding up infinitely many numbers — and when that even makes sense.
A sequence is an ordered list of numbers; a series is their sum. Adding infinitely many things sounds impossible, yet . The central question of this chapter: does an infinite sum settle on a finite value (converge) or blow up (diverge)?
The one you'll use most. When the ratio between terms is constant and less than 1 in size:
Find the sum of the geometric series.
Does the series converge or diverge?
| Test | When to use |
|---|---|
| nth-term test | If terms don't → 0, the series diverges. (Quick disqualifier.) |
| Geometric | Σ ar ⁿ converges iff |r| < 1, to a/(1−r). |
| p-series | Σ 1/nᵖ converges iff p > 1. (So Σ1/n diverges, Σ1/n² converges.) |
| Ratio test | If lim |aₙ₊₁/aₙ| < 1, converges. Great for factorials & powers. |
| Comparison / Integral | Compare to a known series, or to an improper integral. |
The grand finale: represent a function as an infinite polynomial. The Taylor series of centered at 0 (its Maclaurin series) is
A polynomial — the simplest kind of function — can imitate any smooth curve. Each new term widens the interval where the dashed approximation matches. This is how calculators actually compute sin, cos, and eˣ.
A chip can only add and multiply — it can't "take a sine." So it evaluates a few terms of a Taylor (or related) series. Series turn transcendental functions into arithmetic, which is why they underpin everything from graphing calculators to physics engines.
Curves that loop, cross, and spiral — beyond y = f(x).
Not every curve passes the vertical-line test. A circle, a figure-eight, the path of a planet — none can be written as a single . Two new coordinate systems set them free.
Let a parameter (often time) drive both coordinates independently: , . As advances, the point draws the curve.
Instead of left/right and up/down, locate a point by its distance from the origin and angle . The dictionary between the two systems:
Equations like (a rose) or (a cardioid) are trivial in polar but monstrous in –. Try them in the picker above.
Polar area is swept out in pie-slice wedges, not rectangles, so the formula uses : .
Parametric equations describe projectile motion and animation paths; polar coordinates describe radar, spirals in nature, antenna radiation patterns, and anything with rotational symmetry.
Equations whose unknown is a function — the language of science.
A differential equation relates a function to its own derivatives. Instead of solving for a number, you solve for a whole function. They model essentially every changing system in physics, biology, engineering, and finance.
A differential equation describes a curve by its slope. The red curve follows those slopes step by step — that's Euler's method, the simplest numerical ODE solver.
The most solvable type — get all the 's on one side, all the 's on the other, then integrate both sides.
This single equation — "rate of change is proportional to amount" — governs exponential growth and decay.
Compound interest, radioactive decay, unchecked populations.
Population with a carrying capacity M; the S-curve.
An object relaxing to ambient temperature Tₐ.
Differential equations are where derivatives and integrals join forces to predict the future: the trajectory of a rocket, the spread of a disease (the SIR model), the charge in a circuit, the price of an option (Black–Scholes). If calculus has a "final boss," this is it.
Calculus in 3D and beyond: surfaces, partials, gradients, and double integrals.
So far every function took one input. Reality usually has more: temperature depends on latitude and longitude; profit depends on price and volume. A function describes a surface floating above the plane. Calculus III extends derivatives and integrals to these surfaces.
To differentiate a surface, pick a direction. The partial derivative measures the slope as you walk in the x-direction, holding fixed — and vice versa.
a saddle — neither max nor min at the origin. Red is high, blue is low.
A partial derivative just freezes one variable and differentiates in the other — the slope of the surface along that compass direction.
Find the partial derivative with respect to x (treat y as constant).
Bundle the partials into a vector and you get the gradient — it points in the direction of fastest increase:
Just as a single integral sums strips to get area, a double integral sums tiny columns to get the volume under a surface:
You integrate inside-out: do the inner integral (treating the outer variable as constant), then the outer one.
Gradients power gradient descent (all of deep learning). Partial derivatives drive thermodynamics and fluid flow. Multiple integrals compute mass, center of mass, and probability over regions. Vector calculus (div, curl, and the theorems of Green, Stokes, and Gauss) is the mathematics of electromagnetism.
Not just exam fuel — the operating system of modern science and tech.
Whenever something changes or accumulates, calculus is nearby. A tour of where it's quietly running the show:
Position, velocity, and acceleration are a derivative chain. Integrate force to get work, acceleration to get velocity.
Marginal cost and revenue are derivatives of totals; consumer & producer surplus are areas (integrals).
Drug concentration decays exponentially; dosing schedules solve a differential equation to stay in the safe band.
Populations follow logistic growth toward a carrying capacity — a classic differential equation.
Fourier analysis (built on integrals) decomposes sound and images into frequencies — the basis of MP3 and JPEG.
Probabilities are areas under density curves; models train by following gradients downhill (below).
Every neural network — every chatbot, image generator, and recommendation engine — is trained by gradient descent. The model's error is a function of millions of parameters; training means rolling downhill on that error surface by repeatedly stepping against the gradient. Here it is in 1D:
The ball always moves opposite the slope, so it rolls downhill to a minimum. Too big a learning rate overshoots and bounces; start on different sides to see it settle into different local minima — the central challenge of training neural networks.
The derivative you learned as "slope of a tangent line" is the exact tool that lets a computer improve itself. Calculus isn't a relic — it's the engine under the hood of the AI you're talking to right now.
Every rule on this page, in one scannable place. Bookmark it.
Unlimited randomized problems across every topic — with full solutions.
Practice is where calculus actually sticks. Every generator below produces fresh problems forever; each one grades your answer (algebraically equivalent forms are accepted) and reveals a step-by-step solution. Your running totals:
The full firehose — any topic, any difficulty. Great for review or exam prep.
Evaluate the limit.
Evaluate the limit.
Differentiate using the power rule.
Find the equation of the tangent line (solve for y).
Find the indefinite integral (don't forget + C).
Find the sum of the geometric series.
Don't peek at the solution first. Attempt it, check, and only then reveal the steps — that retrieval effort is what builds durable understanding. Aim for a streak of 5 in each topic before moving on.
The questions every calculus learner asks.
Every key term, defined in one line.