Calculus guides
Master derivatives, integrals, limits, and advanced calculus concepts with step-by-step solutions.
48 guides · each with a free step-by-step calculator
Calculus Solver: Master Derivatives & Integrals with Step-by-Step Solutions
Your car's speedometer is doing calculus. It shows how fast your position is changing right now, not your average for the trip. Calculus is simply the maths of change and accumulation. In this guide you'll learn the three big ideas, limits, derivatives and integrals, in plain English with simple examples.
Derivative Calculator Guide: Master the Rules of Differentiation
How steep is a curve at one single point? A straight line has one slope everywhere, but a curve changes its steepness at every step. The derivative answers that question exactly. In this guide you'll learn the four rules that handle almost every derivative: power, product, quotient and chain.
Integral Calculator Guide: Understanding Definite and Indefinite Integrals
If you know how fast water flows into a tank every second, how do you find how much is in it after an hour? You add up all the tiny amounts, and that is exactly what an integral does. In this guide you'll learn definite and indefinite integrals, the power rule for integration, u-substitution and integration by parts.
Limits and Continuity: The Foundation of Calculus
What is 0 divided by 0? It has no answer, yet calculus runs into it all the time. Limits are the clever idea that gets around it: instead of asking what happens AT a point, we ask what happens as we get closer and closer. In this guide you'll learn limits, continuity, the squeeze theorem and L'Hôpital's rule.
Power Rule for Derivatives Explained Simply
What is the derivative of x¹⁰⁰? It sounds scary, but you can answer it in two seconds: 100x⁹⁹. That is the power rule, the first and most-used shortcut in calculus. By the end of this guide you'll use it on squares, roots and negative powers without a second thought.
How to Solve a Definite Integral (Power Rule)
How do you measure the area of a shape with a curved edge, where no ruler or rectangle formula fits? A definite integral does exactly that. In this guide you'll learn to find the area under curves like x² and x³ in four short steps.
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What Is a Partial Derivative? Simple Guide
If you make a box a little wider but keep its height the same, how much extra space do you get? When a formula depends on two things at once, a partial derivative answers exactly that. This guide shows you how to find one using rules you already know.
Chain Rule Explained Simply (With Examples)
Why is the derivative of (3x)² equal to 18x and not just 6x? The missing 3 is the chain rule at work, and it is the rule students forget most often. This guide shows you how to spot a function inside a function and differentiate it correctly every time.
Product Rule for Derivatives (Easy Guide)
If the derivative of x² is 2x and the derivative of x³ is 3x², is the derivative of x² × x³ just 2x × 3x²? No, and that wrong guess is one of the most common calculus errors. This guide shows you the product rule, why it works, and how to use it in five quick steps.
Quotient Rule for Derivatives Made Easy
Can you remember a formula by singing it? Thousands of students learn the quotient rule with one silly rhyme: “low d-high minus high d-low, over low squared”. This guide explains what it means, why the minus sign is there, and how to use it without mixing up the order.
Derivative of sin(x) and sin(ax) Explained
The derivative of sin(x) is cos(x), and the derivative of sin(ax) is a·cos(ax). But why should the slope of one wave be another wave? This guide shows you the picture behind the rule, then walks through easy examples you can check in seconds.
Derivative of cos(x) and cos(ax), with Examples
The derivative of cos(ax) is −a·sin(ax), so plain cos(x) gives −sin(x), but where does that sneaky minus sign come from? It trips up more students than any other trig rule. This guide shows you why it’s there and walks through easy examples.
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Derivative of e^x and e^(ax) Explained
Is there a function that is its own derivative? Yes: eˣ. Its slope at every point is exactly equal to its height, which makes it one of the most important functions in science. This guide shows you why, and how to differentiate e^(ax) in one step.
Derivative of ln(x) and ln(ax): Easy Guide
The derivative of ln(ax) is simply 1/x, so ln(2x), ln(5x) and ln(x) all have exactly the same derivative. Surprised that the a vanishes? Most students are. This guide shows you why it happens, and when a +b inside changes the answer.
Implicit Differentiation Explained Simply
How do you find the slope of a circle, when you can’t even write it as “y = …” without a ± sign? Implicit differentiation lets you skip that problem completely. This guide shows you the method on the circle x² + y² = r², one clear step at a time.
How to Find Critical Points (Step-by-Step)
Where does a profit curve peak, or a cost curve bottom out? At the exact spots where the graph goes flat. Those spots are called critical points, and finding them is the first step in almost every “best value” problem. This guide shows you how in four steps.
Higher-Order Derivatives: Second & Third
Your car’s speedometer shows a first derivative. The push you feel back into your seat when you speed up is a second derivative. Engineers even have a name for the third one: “jerk”. This guide shows you what each level means and how to find them for powers of x.
L'Hôpital's Rule Explained (With Examples)
What is 0 divided by 0? On its own, it has no answer. But a limit that looks like 0/0 often has a perfectly good one, and L’Hôpital’s Rule helps you find it by taking two quick derivatives. This guide shows you when the rule works and when it doesn’t.
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Taylor Series Explained: e^x Step by Step
How does a calculator find e^0.5 when it can really only add and multiply? One common way is a Taylor series, which builds the answer from simple powers of x. Just 10 terms give e correct to five decimal places. This guide shows you how it works, step by step.
Mean Value Theorem Explained Simply
If you drive 120 km in 2 hours, was your speedometer ever at exactly 60 km/h? Yes, at least once, guaranteed. That simple idea is the Mean Value Theorem. This guide explains what it says, when it works, and how to find the point c where it happens.
Integral of sin(ax) and cos(ax) Explained
Is the integral of sin x equal to cos x or −cos x? Even strong students pause on this one. The integral of sin(ax) is −cos(ax)/a + C and of cos(ax) is sin(ax)/a + C. This guide shows where they come from and a 5-second check so you never mix up the sign again.
Integration by Parts: Formula and Examples
What do you do when an integral has two different kinds of function multiplied together, like x × eˣ? No basic rule works directly. Integration by parts swaps that hard integral for an easier one. This guide shows you the formula, a simple way to choose u, and worked examples.
U-Substitution for Integrals, Step by Step
What if you could turn a messy integral into one you already know, just by renaming part of it? That is u-substitution, the most-used trick in integration. This guide shows you how to choose u, what to do with dx, and how to handle the limits.
What Is a Riemann Sum? Left, Right, Midpoint
How can you find the area of a curved shape using only rectangles? Fill it with thin strips, add them up, and you are surprisingly close. That is a Riemann sum, the idea behind every integral. This guide shows you the left, right and midpoint methods with easy examples.
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Trapezoidal Rule: How to Estimate Integrals
How do surveyors find the area of a field with a curved river along one side, using only a tape measure? They often use the trapezoidal rule: join measured points with straight lines and add up the trapezoids. This guide shows you the formula and a full worked example.
Simpson's Rule Explained with a Worked Example
How do you find the area under a curve when no formula will give it to you? Simpson's Rule fits tiny parabolas to the graph and adds up their areas, and with just 4 strips it can land within 0.3% of the true answer. Here's how the 1-4-2-4-1 pattern works and how to use it.
Arc Length of a Curve: Formula and Examples
How long is a curved path, a hanging cable or a bent wire? A ruler can't follow a curve, but calculus can: it chops the curve into tiny straight pieces and adds them up. In this guide you'll learn the arc length formula, where it comes from, and how to use it.
Surface Area of Revolution: Formula and Steps
How much metal goes into a satellite dish, or glass into a curved lamp shade? Spin a curve around an axis and it sweeps out a 3D surface. This guide shows how to find the area of that surface by adding up thin rings, with a clear example.
Improper Integrals: Convergence and the p-Test
Can a region that stretches on forever have a finite area? Surprisingly, yes. The area under 1/x² from 1 to infinity is exactly 1, but the area under 1/x never stops growing. This guide shows you how to tell the difference with one simple test.
How to Find the Area Between Two Curves
How much space is trapped between two winding graphs? Think of a strip of land between two curved roads. You find its area by subtracting the lower curve from the upper one and adding up thin strips. This guide walks you through it with a leaf-shaped example.
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Disk Method for Volume of Revolution Explained
How much does a curved bowl, vase or bottle hold? If you can draw its side as a curve, you can find its volume. The disk method slices the shape into thin coins and adds them up. This guide shows the idea, the formula and a full example.
Shell Method for Volume Explained Simply
What if you spin a region around the y-axis instead of the x-axis? Slicing into coins gets awkward, but there's a neater trick: build the solid from thin nested tubes, like the layers of an onion or a roll of paper. This guide explains the shell method and when to use it.
What Is the Gradient Vector? Meaning and Examples
Standing on a hillside in fog, which way is straight uphill? The gradient answers that for any function of two variables. It's an arrow pointing where the function rises fastest, and its length tells you how steep. This guide shows how to find it in a few steps.
Directional Derivative: Formula and Examples
Walking north-east up a hill, how steep is your path? A partial derivative only tells you the slope going exactly east or exactly north. The directional derivative gives the slope in any direction you choose. This guide shows how, using the gradient and one dot product.
Double Integrals Explained: How to Evaluate Them
How do you find the total mass of a plate that's heavier at one corner, or the volume under a curved roof? A single integral adds along a line; a double integral adds over a whole flat region. This guide shows how to evaluate one, one integral at a time.
What Is the Jacobian? 2×2 Determinant Explained
Why does dA become r dr dθ in polar coordinates? Where does that extra r come from? It's a Jacobian: the number that tells you how much a change of variables stretches or shrinks area. This guide explains what it means and how to compute it in four quick steps.
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What Is Divergence? Vector Field Divergence Explained
Is water spreading out from a point, like a tap, or draining into it, like a plughole? Divergence measures exactly that for any flow. This guide explains what the number means, how to compute it in two quick derivatives, and what its sign tells you.
What Is Curl? 2D Curl of a Vector Field Explained
Drop a tiny paddle wheel into a flowing river. Will it spin, and which way? That's what curl measures. This guide explains the idea with a simple picture, gives the 2D formula, and shows how to read the sign of your answer.
How to Solve Separable Differential Equations
Why do rabbit populations, savings and cooling coffee all follow the same curve? Because each one changes at a rate that depends on how much there is. This guide shows how to solve that kind of equation by separating the variables, with a full rabbit example.
Integrating Factor Method for First-Order ODEs
Fill a leaky bucket from a tap and the water level rises fast, then slows, then settles. That story is a first-order linear differential equation. This guide shows how the integrating factor solves it, and how to spot the steady level straight away.
Second-Order Homogeneous ODEs: Characteristic Equation
Why does a car bounce once and settle, while a guitar string rings for seconds? Both follow ay'' + by' + cy = 0. One clever guess, y = eʳˣ, turns this calculus problem into a quadratic equation. This guide shows how, and what each type of root means.
Euler's Method Explained with a Step-by-Step Example
How does a computer solve an equation it can't solve exactly? It takes tiny straight steps, checking the direction after each one, like walking through fog with a compass. That's Euler's method. This guide shows how it works, with a table you can follow by hand.
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Exponential Growth and Decay: Formula and Examples
Why does a colony of bacteria seem to explode overnight, while a medicine fades from your blood more and more slowly? Both follow the same formula, P = P₀eᵏᵗ. This guide explains it, shows how to find doubling time and half-life, and works through a lab example.
How to Find the Limit of a Polynomial
What value is a function heading towards as x gets closer and closer to a number? That's a limit, the idea all of calculus is built on. For polynomials the answer is refreshingly simple: just plug the number in. This guide explains why, and when that shortcut fails.
Squeeze Theorem Explained: Why sin(x)/x → 1
What is sin(x)/x when x is 0? Plugging in gives 0/0, which tells you nothing. Yet the answer is exactly 1, and the squeeze theorem proves it by trapping the function between two simpler ones. This guide explains the idea with an everyday picture and a table you can check.
How to Check Continuity of a Function at a Point
Can you draw a graph through a point without lifting your pencil? If yes, the function is continuous there. Calculus turns that simple test into three checks you can do on paper. This guide explains them, shows the three kinds of breaks, and works through an example.
Fourier Series Explained: Square Wave Coefficients
How can a sharp-cornered square wave be made from smooth sine waves? It sounds impossible, but add enough of them and the corners appear. That's a Fourier series. This guide explains the idea like a musical chord and shows how to find the square wave's coefficients.
Laplace Transform Explained with Steps and Examples
What if you could turn a hard calculus problem into simple algebra? That's what the Laplace transform does: derivatives become multiplication, and differential equations become equations you can solve in a few lines. This guide explains the idea and works out L{e^(at)} = 1/(s − a) step by step.
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