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Learn Calculus in Plain English
Twelve short chapters, no jargon. Every idea is explained with simple words, a picture, and one worked example. Start at Chapter 1 or jump anywhere.
Chapter 1
Functions and Graphs
Before calculus makes sense, you need one idea: a function is a machine that turns one number into another number.
Key idea
A function takes an input, follows a rule, and gives back exactly one output.
We write a function like this: f(x) = x^2. Say it out loud as "f of x equals x squared". The letter x is the input. The x^2 part is the rule. Whatever number you put in, you square it and that is your answer.
One input goes in, one output comes out. That is all a function is.
The graph is just a picture of the machine
Put the input across the bottom (the x-axis) and the output up the side (the y-axis). Mark a dot for every input-output pair. Join the dots and you have the graph.
Domain — every input the function is allowed to take.
Range — every output it can actually produce.
Slope — how steep the graph is. Steep means the output changes fast.
Why this matters for calculus
A straight line has the same slope everywhere, so it is easy: slope equals rise over run. A curve is different — its steepness changes at every single point. Measuring the steepness of a curve at one exact point is the whole problem that calculus was invented to solve.
Function value f(x) = x² → f(3) = 9, f(-2) = 4
Slope of a line m = (y₂ − y₁) / (x₂ − x₁)
Worked example
Question: For f(x) = 2x + 1, find f(4) and the slope.
Step 1. Replace every x with 4: f(4) = 2(4) + 1 = 9.
Step 2. This is a straight line in the form y = mx + b, so the slope is the number in front of x, which is 2. It is 2 everywhere on the line.
Remember this
A function gives exactly one output for each input.
The graph is a picture of all those input-output pairs.
Lines have one fixed slope; curves change slope constantly.
Calculus exists to measure change on curves.
Chapter 2
Limits
A limit asks a simple question: where is this function heading, even if it never actually gets there?
Key idea
The limit is the value a function gets closer and closer to as the input gets closer and closer to some number.
Imagine walking toward a door, and each step you take covers half the remaining distance. You never touch the door — but anyone watching knows exactly where you are heading. The door is the limit.
Here is the classic case. Take f(x) = (x^2 - 4)/(x - 2). If you plug in x = 2 you get 0/0, which is meaningless. But look at what happens near 2:
At x = 1.9 the answer is 3.9
At x = 1.99 the answer is 3.99
At x = 2.01 the answer is 4.01
From both sides the values crowd around 4. So the limit is 4, even though the function itself has a hole at that spot.
Both sides head to the same height, so the limit exists — even though the point itself is missing.
How to actually find a limit
Try substitution first. Put the number in. If you get a normal answer, you are done.
If you get 0/0, the expression can be simplified. Factor the top and bottom, cancel the matching part, then substitute again.
Check both sides. The limit from the left and the limit from the right must agree. If they do not, the limit does not exist.
Limits at infinity
Sometimes we ask what happens when x runs off forever. For a fraction, compare the biggest power on top with the biggest power on the bottom. If the bottom grows faster, the answer heads to 0. If they grow at the same rate, the answer is the ratio of the leading numbers.
Question: Find lim (x^2 - 9)/(x - 3) as x approaches 3.
Step 1. Substitute 3: you get 0/0. Not allowed, so keep going.
Step 2. Factor the top: x^2 - 9 = (x - 3)(x + 3).
Step 3. Cancel the (x - 3) on top and bottom, leaving x + 3.
Step 4. Now substitute: 3 + 3 = 6. The limit is 6.
Remember this
A limit is a destination, not necessarily a stop.
0/0 is a signal to simplify, not an answer.
Left and right must agree or there is no limit.
Limits are the foundation for both derivatives and integrals.
Chapter 3
The Derivative
The derivative is the slope of a curve at one exact point — in other words, how fast something is changing right now.
Key idea
The derivative turns a function into a new function that reports the steepness at every point.
Your car's speedometer is a derivative. The odometer tells you distance. The speedometer tells you how fast that distance is changing at this instant. Calculus calls that instant rate of change the derivative.
How we get there
Pick two points on a curve and draw a straight line through them. That line is called a secant, and its slope is easy to compute. Now slide the second point closer and closer to the first. The secant line pivots until it just grazes the curve at a single point. That final line is the tangent, and its slope is the derivative.
As the second point slides in, the secant line becomes the tangent line.
The definition
Written formally, the derivative is the limit of the secant slope as the gap h shrinks to nothing. In practice nobody computes this by hand for long — you learn the shortcut rules in the next chapter — but this is where they all come from.
Definition f′(x) = limh→0 [ f(x + h) − f(x) ] / h
Same thing, other names f′(x) = dy/dx = y′
Meaning f′(a) = slope of the tangent line at x = a
What the sign tells you
f′(x) > 0 — the function is going up.
f′(x) < 0 — the function is going down.
f′(x) = 0 — the function is momentarily flat: a peak, a valley, or a pause.
Worked example
Question: Find the derivative of f(x) = x^2 from the definition, then find the slope at x = 3.
Step 1.f(x + h) = (x + h)^2 = x^2 + 2xh + h^2.
Step 2. Subtract f(x): 2xh + h^2.
Step 3. Divide by h: 2x + h.
Step 4. Let h shrink to 0: f′(x) = 2x. At x = 3 the slope is 6.
Remember this
Derivative = slope of the tangent = instant rate of change.
It comes from squeezing a secant line into a tangent line.
The derivative of a function is itself a function.
Positive means rising, negative means falling, zero means flat.
Chapter 4
Derivative Rules
A handful of shortcuts replace the limit definition forever. Learn these and you can differentiate almost anything.
Key idea
Every complicated function is built from simple pieces. Each rule tells you how to handle one type of building block.
The core rules
Constant: the derivative of any plain number is 0. Nothing is changing.
Power rule: bring the exponent down in front, then subtract one from it.
Sum rule: differentiate each piece separately and add the results.
Product rule: two things multiplied need both terms — first times derivative of second, plus second times derivative of first.
Quotient rule: for a fraction, use the "low d-high minus high d-low, over low squared" pattern.
Chain rule: for a function inside another function, multiply the outside derivative by the inside derivative.
The chain rule: peel the function like an onion and multiply the slopes together.
Constant d/dx (c) = 0
Power d/dx (xⁿ) = n·xⁿ⁻¹
Product (uv)′ = u′v + uv′
Quotient (u/v)′ = (u′v − uv′) / v²
Chain d/dx f(g(x)) = f′(g(x)) · g′(x)
Derivatives worth memorising
d/dx (sin x) = cos x | d/dx (cos x) = −sin x
d/dx (eˣ) = eˣ | d/dx (ln x) = 1/x
d/dx (tan x) = sec² x | d/dx (√x) = 1 / (2√x)
Worked example
Question: Differentiate sin(x^2).
Step 1. Spot the layers. The outer function is sin(...). The inner function is x^2.
Step 2. Differentiate the outside, leaving the inside alone: cos(x^2).
Step 3. Differentiate the inside: 2x.
Step 4. Multiply them: 2x·cos(x^2).
Remember this
Power rule: drop the power in front, then reduce it by one.
Products and quotients each have their own pattern — do not just multiply derivatives.
Chain rule is the one students forget most. Always ask "is something inside something else?"
Memorise the six standard derivatives; they appear constantly.
Chapter 5
Using Derivatives
This is where derivatives earn their keep: finding the best possible answer, the highest point, the fastest route.
Key idea
Wherever a smooth curve reaches a peak or a valley, it flattens out for an instant — so the derivative is zero there.
Finding highs and lows
A point where f′(x) = 0 is called a critical point. It is a candidate for a maximum or a minimum. To decide which one it is, look at what happens around it:
Slope changes from positive to negative → you just went over a peak (local maximum).
Slope changes from negative to positive → you just passed through a valley (local minimum).
Slope does not change sign → it is a flat spot but neither a peak nor a valley.
At the top of a hill and the bottom of a dip, the tangent line is perfectly flat.
The second derivative
Differentiate twice and you get f″(x), which describes the bend of the curve. If the first derivative is speed, the second derivative is acceleration.
f″(x) > 0 — curve bends upward like a cup. A critical point here is a minimum.
f″(x) < 0 — curve bends downward like a cap. A critical point here is a maximum.
f″(x) = 0 — the bend may be switching direction. This is an inflection point.
Real jobs derivatives do
Optimisation: the cheapest packaging, the largest area for a fixed fence, the maximum profit.
Related rates: a ladder slides down a wall — how fast does the bottom move?
Motion: position → velocity → acceleration, each one a derivative of the last.
Critical points solve f′(x) = 0
Second derivative test f″(a) > 0 → minimum · f″(a) < 0 → maximum
Motion s(t) → v(t) = s′(t) → a(t) = v′(t)
Worked example
Question: Find the minimum of f(x) = x^2 - 4x + 7.
Step 1. Differentiate: f′(x) = 2x - 4.
Step 2. Set it to zero: 2x - 4 = 0, so x = 2.
Step 3. Check the bend: f″(x) = 2, which is positive, so it is a minimum.
Step 4. The lowest value is f(2) = 4 - 8 + 7 = 3.
Remember this
Set the derivative to zero to find candidates for peaks and valleys.
The second derivative tells you which one you found.
Optimisation problems are just critical-point problems in disguise.
Always check the endpoints too when a range is given.
Chapter 6
The Integral
If the derivative chops things apart, the integral glues them back together. It measures total accumulation — usually area.
Key idea
An integral adds up infinitely many infinitely thin slices to get an exact total.
Suppose you want the area under a curve. You cannot use a rectangle formula because the top is not flat. So you cheat: cover the region with thin rectangles, add up their areas, then make the rectangles thinner and thinner. The number those sums close in on is the integral.
Each strip is height × width. Add them all and shrink the width to zero — that is the integral.
Two kinds of integral
Indefinite integral — asks "which function has this derivative?" The answer is a family of functions, so we add + C at the end.
Definite integral — has a start and an end value, and gives a single number: the accumulated total between those bounds.
The Fundamental Theorem
This is the big one, and it is beautifully simple: differentiation and integration undo each other. To find a definite integral you do not need any rectangles at all. Just find an antiderivative, plug in the top bound, plug in the bottom bound, and subtract.
Power rule ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)
Fundamental theorem ∫ab f(x) dx = F(b) − F(a), where F′ = f
Must-know ∫ 1/x dx = ln|x| + C · ∫ eˣ dx = eˣ + C
Trig ∫ sin x dx = −cos x + C · ∫ cos x dx = sin x + C
Worked example
Question: Find the area under f(x) = x^2 from 0 to 3.
Step 1. Find the antiderivative: F(x) = x^3/3.
Step 2. Plug in the top bound: F(3) = 27/3 = 9.
Step 3. Plug in the bottom bound: F(0) = 0.
Step 4. Subtract: the area is 9.
Remember this
An integral is a total built from tiny pieces.
Indefinite integrals need + C; definite integrals give a number.
The power rule for integrals is the reverse of the one for derivatives.
The Fundamental Theorem links the two halves of calculus.
Chapter 7
Integration Techniques
Integrals do not have a chain rule or a product rule. Instead we have tricks that reshape a hard integral into an easy one.
Key idea
You are not solving the integral directly — you are rewriting it until it becomes one you already recognise.
1. Substitution (the reverse chain rule)
Use this when you can spot a function and its derivative sitting in the same integral. Rename the inner function as u, swap everything over, and the integral usually collapses into something trivial.
Substitution renames the messy inside piece so the whole integral becomes familiar.
2. Integration by parts (the reverse product rule)
Use this when two unrelated things are multiplied, like x·e^x or x·ln(x). Choose one part to differentiate and one to integrate. A handy order for picking u is LIATE: Logarithm, Inverse trig, Algebra, Trig, Exponential.
3. Partial fractions
Use this when you have a fraction with a factorable bottom. Split the one ugly fraction into two or three simple ones, then integrate each with the natural log rule.
4. Trigonometric rewriting
Use identities to lower powers before integrating. For example, replace sin^2(x) with (1 - cos(2x))/2, which is easy to integrate term by term.
Substitution ∫ f(g(x))·g′(x) dx = ∫ f(u) du
By parts ∫ u dv = uv − ∫ v du
Log rule ∫ f′(x)/f(x) dx = ln|f(x)| + C
Worked example
Question: Find ∫ 2x·cos(x^2) dx.
Step 1. Notice that x^2 is inside the cosine, and its derivative 2x is right there in front. That is the signal to substitute.
Step 2. Let u = x^2, so du = 2x dx.
Step 3. The integral becomes ∫ cos(u) du = sin(u) + C.
Step 4. Swap back: the answer is sin(x^2) + C.
Remember this
Substitution: look for a function and its derivative together.
By parts: use it when two different types of function are multiplied.
Partial fractions: use it when the denominator factors.
If one method stalls, try another — integration is pattern matching.
Chapter 8
Using Integrals
Area is only the beginning. The same idea gives you volumes, averages, distances and totals of every kind.
Key idea
Whenever you can slice a quantity into thin pieces and add them up, an integral will give you the exact total.
Area between two curves
Take the top curve, subtract the bottom curve, and integrate the difference across the region. The gap between them is the height of each thin strip.
Height of each strip = top minus bottom. Integrate that height across the region.
Volume of a solid of revolution
Spin a curve around an axis and it sweeps out a solid. Slice that solid into thin discs. Each disc is a circle of radius f(x) and thickness dx, so its volume is π·f(x)²·dx. Integrate to add up all the discs.
Average value
The average of a list is the sum divided by how many items. The average of a function is the integral divided by the width of the interval — same idea, continuous version.
Distance from speed
If you know how fast something moved at every instant, integrating the speed gives total distance travelled. This is exactly why an integral is called an accumulation.
Area between A = ∫ab [ f(x) − g(x) ] dx
Disc volume V = π ∫ab [ f(x) ]² dx
Average value favg = (1/(b−a)) ∫ab f(x) dx
Distance d = ∫ab |v(t)| dt
Worked example
Question: Find the average value of f(x) = x^2 between 0 and 3.
Step 1. Integrate: ∫₀³ x^2 dx = 9 (from the last chapter).
Step 2. The width of the interval is 3 - 0 = 3.
Step 3. Divide: 9 / 3 = 3. The average value is 3.
Remember this
Area between curves is always top minus bottom.
Volumes come from slicing a solid into discs or shells.
Average value spreads the total evenly across the interval.
Integrating a rate always gives you the total amount.
Chapter 9
Sequences and Series
What happens when you add up an endless list of numbers? Sometimes the total settles down. Sometimes it runs away forever.
Key idea
A sequence is a list of numbers. A series is what you get when you add that list up.
A sequence is just numbers in order: 1, 1/2, 1/4, 1/8, and so on. We ask whether the terms settle toward a single value. These ones head to 0, so the sequence converges.
A series adds them: 1 + 1/2 + 1/4 + 1/8 + … Surprisingly, this endless sum has a finite answer of exactly 2. Each new term is half of what remains, so you creep toward 2 without ever passing it.
If the running totals settle on a number, the series converges. If they keep climbing, it diverges.
How to tell if a series converges
nth term test: if the terms do not shrink to 0, the series definitely diverges. This is the first thing to check.
Geometric series: if each term is the previous one times r, it converges whenever |r| < 1.
p-series: a sum of 1/n^p converges when p > 1. The famous harmonic series (p = 1) diverges, even though its terms shrink.
Ratio test: divide one term by the previous one and take the limit. Below 1 converges, above 1 diverges.
Geometric sum a + ar + ar² + … = a / (1 − r), for |r| < 1
p-series Σ 1/nᵖ converges ⟺ p > 1
Ratio test L = lim |an+1 / an| · L < 1 converges
Worked example
Question: Does 1 + 1/2 + 1/4 + 1/8 + ... have a finite total?
Step 1. Each term is the one before it multiplied by 1/2, so this is geometric with a = 1 and r = 1/2.
Step 2. Since |r| = 0.5 < 1, it converges.
Step 3. Use the formula: 1 / (1 - 0.5) = 2. The total is exactly 2.
Remember this
Sequence = a list. Series = the sum of that list.
Terms shrinking to zero is necessary but not enough for convergence.
Geometric series are the ones you can total exactly.
The harmonic series is the classic trap: shrinking terms, infinite sum.
Chapter 10
Taylor and Maclaurin Series
Any smooth curve can be imitated by a polynomial. Add more terms and the imitation gets better and better.
Key idea
Match the value, then the slope, then the bend, then the next bend — and your polynomial becomes indistinguishable from the real function.
Your calculator does not "know" sin(0.3). It uses a polynomial that behaves almost exactly like sine near that point, because polynomials only need addition and multiplication. That polynomial is a Taylor series.
When the series is built around x = 0, it gets the special name Maclaurin series. That is the version you meet most often.
A straight line matches the curve for a moment. Higher-degree polynomials hug it much further.
How it is built
Each term uses one more derivative than the last, divided by a factorial to keep the sizes under control. The first term matches the height, the second matches the slope, the third matches the curvature, and so on.
Question: Build the Maclaurin series for e^x up to the x³ term.
Step 1. Every derivative of e^x is e^x, and at x = 0 each one equals 1.
Step 2. Feed those into the formula: 1 + 1·x + 1·x^2/2! + 1·x^3/3!.
Step 3. Simplify: 1 + x + x^2/2 + x^3/6.
Check: at x = 0.1 this gives 1.10517, and the true value of e^0.1 is 1.10517. Close enough for four terms.
Remember this
Taylor series turn hard functions into polynomials.
Maclaurin is simply a Taylor series centred at zero.
More terms means a better fit over a wider range.
The series for eˣ, sin x and cos x are worth memorising.
Chapter 11
Partial Derivatives
Real problems depend on several things at once. A partial derivative asks how the answer changes when you move just one of them.
Key idea
To take a partial derivative, treat every other variable as if it were a plain number.
Imagine standing on a hill. Your height depends on two things: how far east you are and how far north you are, so we write f(x, y). Now ask two separate questions:
If I take one step east, how much does my height change? That is ∂f/∂x.
If I take one step north, how much does my height change? That is ∂f/∂y.
The curly ∂ symbol just means "partial". It is a reminder that other variables are being held still.
Two directions, two slopes. Each partial derivative holds the other direction frozen.
Second partials and the gradient
You can differentiate twice, and even mix the variables: differentiate by x first, then by y. For well-behaved functions the order does not matter — you get the same answer either way.
Collect both first partials into a pair and you get the gradient. It points in the steepest uphill direction, which is exactly why machine learning uses gradient descent to walk downhill toward a minimum.
Partial ∂f/∂x — differentiate with respect to x, treat y as a constant
Gradient ∇f = ( ∂f/∂x , ∂f/∂y )
Mixed partials ∂²f/∂x∂y = ∂²f/∂y∂x
Worked example
Question: For f(x, y) = x^2·y + 3y, find both partial derivatives.
Step 1. For ∂f/∂x, pretend y is a number. The derivative of x^2·y is 2xy, and 3y is a constant so it vanishes. Answer: 2xy.
Step 2. For ∂f/∂y, pretend x is a number. The derivative of x^2·y is x^2, and the derivative of 3y is 3. Answer: x^2 + 3.
Remember this
Freeze the other variables and differentiate as usual.
The ∂ symbol signals that more than one variable exists.
The gradient points straight uphill.
Mixed partials are equal for smooth functions.
Chapter 12
Differential Equations
An equation that contains a derivative. Instead of solving for a number, you solve for a whole function.
Key idea
A differential equation describes how something changes. Solving it means finding the function that behaves that way.
Ordinary equations ask "what number is x?" Differential equations ask "what function behaves like this?" For example, y′ = y says: find a function whose rate of growth always equals its current size. Population, compound interest and radioactive decay all speak this language.
The equation dictates a slope everywhere. A solution is any curve that follows those slopes.
Separable equations
The friendliest type. If you can shuffle all the y terms to one side and all the x terms to the other, just integrate both sides and you are done.
Initial conditions
Integration always leaves a constant C, so a differential equation has infinitely many solutions — a whole family of parallel curves. One extra fact, such as "the population started at 500", pins down C and picks out the single curve you want.
This free online calculus calculator computes derivatives, integrals, limits, Taylor series, and more — directly in your browser. Enter any expression in x (or another variable) and choose an operation to get an instant symbolic or numeric result with step hints.
It is also a complete learning resource. The Learn tab holds a free twelve-chapter calculus course written in plain English, each chapter with its own infographic, key formulas and a worked example. The Practice tab has a quiz for every chapter plus a timed final exam, and the Formulas tab collects every formula on a single revision page.
What calculus operations does this calculator support?
First and second derivatives, partial derivatives, definite and indefinite integrals, double integrals, limits, Taylor series, critical points, root finding, evaluation, and simplification.
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