Calculus,
made easy.

Calculus has a reputation for being terrifying, mostly because mathematicians like to use intimidating symbols to guard their secrets. Thompson's first goal is to deliver you from these "preliminary terrors."

"What one fool can do, another can."

The entire subject hinges on just two symbols that you need to demystify right now.

The first is d, which merely means "a little bit of." Thus dx means a little bit of x; or du means a little bit of u. Ordinary mathematicians think it is more polite to say "an element of," just as you would say "an element of regret" instead of "a little bit of regret."

The second is ∫, which is merely a long S, and means "the sum of." Thus ∫ dx means the sum of all the little bits of x.

That is all. If you can understand that d breaks things apart into little bits, and ∫ glues them back together, you are ready to learn calculus.

Calculus rests on a single, slippery idea: things can be indefinitely small without being nothing. Thompson calls these little bits dx and dy - where the d simply means "a little bit of."

"In mathematical language, infinitely small means nothing more nor less than indefinitely small."

A little bit of x is written dx. But Thompson's greatest lesson is about degrees of smallness. If dx is a small fraction, like 1/60, then what is dx² (a small bit of a small bit)? It is 1/3600.

A smallness of the second degree (dx² or dx·dy) is microscopically tiny compared to a smallness of the first degree (dx). When we eventually let these bits shrink toward zero, the second-degree bits vanish entirely, leaving only the first-degree bits to form our rate of change.

Drag the slider to shrink dx. Watch how dx² crashes toward zero exponentially faster.

Here is the central question of all calculus: if y depends on x, how fast does y change when x changes? Not by how much - by how fast, per unit of x.

Thompson answers it with a square. Suppose the side is x = 3, so the area is y = x² = 9. Now grow the side by a tiny bit dx. The new area is:

(x + dx)² = x² + 2x · dx + dx²

Three pieces appear: the original square, two thin strips (2x · dx), and a tiny corner (dx²). When dx is very small, the corner is negligibly small compared to the strips - a small times a small is a very small indeed.

Drop the corner. The change in area is just 2x · dx. Divide both sides by dx and you have the rate of change: dy/dx = 2x. The power rule didn't come from nowhere - it came from this square.

This procedure - growing x by dx, finding the resulting change dy, dividing, then letting dx vanish - is called differentiation. The result, dy/dx, is the derivative. Every shortcut and rule in calculus is just a faster way of doing exactly this.

The simplest functions are powers of x: x², x³, x⁵. Thompson shows they all differentiate by one elegant pattern - the power "comes down" and multiplies, then the exponent drops by one:

"If y = xⁿ, then dy/dx = n · xⁿ⁻¹."

So x³ differentiates to 3x². The rate of change of a cube grows as a square. A straight line (n = 1) has a constant derivative - it changes at the same pace everywhere, which makes sense.

Adjust n with the slider. The amber curve is the function; the teal dashed curve is its derivative. Notice how steeper functions have steeper derivatives.

What happens when we differentiate a constant? Think about what a derivative is: it's a measure of how fast something is changing.

But a constant, by definition, never changes. If c = 5, it will always be 5. Its rate of change is absolutely zero.

"The differential of any constant is zero."

Therefore, if you have an equation like y = x² + c, when you differentiate it, the c simply vanishes: dy/dx = 2x.

Geometrically, adding a constant just moves the curve up or down. But it doesn't change the steepness (the slope) at any given x coordinate.

In the interactive below, you can drag your mouse across the grid to move the point and see the slope (just like Chapter X). But now, you can also use the slider to change the constant c. Watch how the curve moves up and down, but the tangent's slope at any given x remains exactly the same!

Thompson provides a beautiful geometric intuition for differentiating a product of two variables, u and v. Instead of just memorizing a rule, he asks you to imagine a rectangle with width u and height v.

The area is uv. Now, suppose the rectangle grows a tiny bit: the width grows by du and the height grows by dv.

"The new area is (u + du)(v + dv) = uv + u·dv + v·du + du·dv."

The original area is uv. The growth consists of three new pieces: a strip on the bottom (u·dv), a strip on the side (v·du), and a tiny little rectangle in the corner (du·dv).

But remember the lesson of degrees of smallness! If du and dv are both small, then their product du·dv is a small quantity of the second order. It is so microscopically tiny compared to the strips that we can throw it away.

Adjust the sliders below to shrink du and dv. Watch how the corner piece du·dv shrinks much faster than the strips until it becomes utterly negligible.

If you differentiate a function, you get its rate of change. But what happens if you differentiate that result again?

Thompson calls this Successive Differentiation. If y is your original function, its first derivative is dy/dx (how fast y is changing). Its second derivative is written as d²y/dx², which simply means: how fast is dy/dx changing?

"Let us try the experiment of differentiating a function twice over."

If y = 0.2x³ - 2x, the first derivative is dy/dx = 0.6x² - 2. The second derivative is d²y/dx² = 1.2x. Each time you differentiate, the power rule bumps the exponent down by one.

This has profound physical meaning. If y is position, then dy/dx is velocity (how fast position changes), and d²y/dx² is acceleration (how fast velocity changes).

In the interactive below, you can see all three functions plotted together. Move your mouse along the x-axis to track a point across all three curves simultaneously. Use the buttons to isolate specific curves and see how the slope of one curve becomes the height of the next!

Calculus isn't just about abstract curves; it's the language of the physical universe. When our independent variable is time (t), those successive derivatives take on names you already know.

Let y be the height of a thrown ball. The first derivative with respect to time, dy/dt, tells us how fast the height is changing. This is velocity.

The second derivative, d²y/dt², tells us how fast the velocity is changing. This is acceleration.

"The rate of change of the rate of change of the distance is the acceleration."

If we throw a ball upward with an initial velocity of 40 m/s, under Earth's gravity, its height is y = 40t - 4.9t². By the power rule, its velocity is dy/dt = 40 - 9.8t. Differentiate again, and its acceleration is d²y/dt² = -9.8.

Notice how the acceleration is a constant negative number? Gravity pulls down at exactly the same rate, regardless of whether the ball is flying up, paused at its peak, or falling down! Press Play on the interactive below to see this in motion.

Sometimes you have a function buried inside another function. How do you find the rate of change of the outer one with respect to the innermost variable?

Thompson calls his solution a "useful dodge" - which we now know as the Chain Rule. He explains it with a very intuitive idea: if y depends on u, and u depends on x, their rates of change simply multiply together.

"If y depends on u, and u depends on x, then the rate of variation of y with regard to x is the product of the rate of variation of y with regard to u, and that of u with regard to x."

Think of it like a chain of gears or pulleys. If gear u turns twice as fast as gear x (so du/dx = 2), and gear y turns three times as fast as gear u (so dy/du = 3), then how fast does y turn relative to x? It turns 2 × 3 = 6 times as fast!

Mathematically: dy/dx = (dy/du) × (du/dx). It looks like you're just cancelling the du's as if they were ordinary fractions. And because they represent "a little bit of," you actually can!

Play with the sliders below to adjust the "gear ratios" and watch how the final speed of y is purely a product of the links in the chain.

The derivative has a beautiful geometric interpretation. If you zoom in infinitely close to any smooth curve, it looks perfectly straight - and the derivative gives the slope of that line.

"The slope of the tangent at any point of a curve is the value of dy/dx at that point."

For the parabola y = x², the derivative is 2x. At x = 1 the slope is 2; at x = −1 it's −2; at the very bottom it's exactly zero - the curve is momentarily flat.

Move your mouse (or touch) across the graph. The tangent line pivots and the slope readout updates in real time. The sign of the slope tells you whether the function is rising or falling.

Here is the payoff. When a function reaches a peak or a valley, it is - for an instant - neither rising nor falling. The slope of the tangent is exactly zero.

"When dy/dx = 0, the function is at a maximum or minimum."

This is why engineers, economists, and scientists use calculus daily: to find the best answer to a question. The shortest path, the strongest beam, the cheapest design - all are found by setting the derivative to zero and solving.

The curve below is f(x) = x³ − 3x, whose derivative is f′(x) = 3x² − 3. Setting that to zero: x = ±1. Move along the curve - watch the tangent flatten exactly at those two points, one a peak, one a valley.

How do you tell which is which? The second derivative: if f′′ < 0 the curve bends downward (a maximum); if f′′ > 0 it bends upward (a minimum). Intuition first, rule second.

We know that the first derivative (dy/dx) tells us the slope of a curve. But what does the second derivative (d²y/dx²) tell us geometrically?

Thompson explains that the second derivative tells us the curvature - whether the curve is bending upwards or bending downwards.

"If the second differential coefficient is positive, the curve is concave upwards... If the second differential coefficient is negative, the curve is concave downwards."

Think of a bowl. If d²y/dx² > 0, the curve is shaped like a cup (concave up), holding water. If d²y/dx² < 0, it is shaped like an umbrella (concave down), shedding water.

If d²y/dx² = 0, the curve is momentarily perfectly straight. This is called a point of inflection - the exact moment where the curve switches from bending one way to bending the other.

In the interactive below, trace the curve. Notice the osculating circle (the "kissing circle"). This circle perfectly hugs the curve, visualizing its curvature. When d²y/dx² is positive, the circle sits above the curve. When negative, it hangs below!

In Chapter 13, Thompson introduces a few more clever tricks ("dodges") to make differentiation easier. One of the most conceptually beautiful ones is the rule for inverse functions.

Suppose you know how to find dy/dx (how fast y changes as x changes), but you actually want to know dx/dy (how fast x changes as y changes).

"Just as dx/dy is the reciprocal of dy/dx, so the differential coefficient of x with respect to y is the reciprocal of the differential coefficient of y with respect to x."

In plain English: if a hill rises 2 feet for every 1 foot you walk forward (slope = 2), then you walk forward 1/2 a foot for every 1 foot you rise.

Mathematically: dy/dx = 1 / (dx/dy). Because dy and dx are just very small numbers, you can literally just flip the fraction upside down!

In the interactive below, trace the curve to see the standard slope (dy/dx). Then, click Swap Axes. The entire graph mirrors itself, swapping the x and y axes. Notice how the steepness instantly inverts, visually proving that dx/dy is just the reciprocal of dy/dx!

We now arrive at one of the most important concepts in all of mathematics: the number e.

Thompson calls it "the law of organic growth." In nature, things don't grow by a fixed amount every day. They grow by a percentage of how much is already there. Your money in the bank grows faster the more money you have. A population of bacteria grows faster the more bacteria there are.

"The rate of growth is proportional to the amount already grown."

If we want to find a function where the rate of growth (dy/dx) is exactly equal to its height (y), there is only one mathematical curve in the universe that does this: y = e^x.

If you differentiate e^x, you just get e^x back. It is completely immune to differentiation!

In the interactive below, trace the curve of y = e^x. We measure its Height (the orange line) and we measure its Slope (the blue triangle, where dx = 1 so dy is the slope). Watch the readout below the graph. No matter where you go on the curve, the steepness is exactly equal to the height!

We just saw how e^x models organic growth, where a quantity grows faster the larger it gets. But what about the reverse? What if a quantity shrinks based on how much is left?

Thompson calls this the Die-away Curve. It models a cooling cup of coffee, radioactive decay, or a discharging capacitor. The mathematical formula is y = e^{-x}.

"As the curve descends, its steepness decreases. Its slope is negative, but the magnitude of the downward slope is exactly equal to the height at that instant."

In calculus terms: dy/dx = -y.

This leads to a beautifully simple geometric fact: if you draw a tangent line touching the curve at any point, it will always cross the x-axis exactly 1 unit ahead of where you are!

In the interactive below, trace the die-away curve. The orange line shows your current Height (y). The blue line is your Tangent. Notice how the base of the triangle (dx) is completely locked at 1. The curve gets shallower precisely as it gets shorter, maintaining that perfect proportion all the way to infinity.

So far, we've dealt with algebraic functions (x², x³) and exponential functions (e^x). Now we turn to the functions that go up and down: the trigonometric functions.

If you plot y = sin(x), it starts at 0, climbs to a peak at x = π/2, falls back through 0 at x = π, hits a valley at x = 3π/2, and returns to 0 at x = 2π.

But what is its rate of change? Where is it growing the fastest? Right at the start! At x = 0, it's shooting upwards at a steep 45° angle (slope = 1). But as it reaches the top of the hill at x = π/2, it flattens out perfectly (slope = 0). Then it starts falling, getting steeper and steeper until it's plunging downwards at a 45° angle (slope = -1) when crossing the axis at x = π.

"The steepness of the sine curve is itself a cosine curve."

Mathematically: If y = sin(x), then dy/dx = cos(x).

And if you start with the cosine curve? It starts at a flat peak (slope = 0), and then falls. Because it starts by going down, its derivative is negative: If y = cos(x), then dy/dx = -sin(x).

Use the interactive below to see this beautiful relationship. Trace along the base curve in orange, and watch how its slope perfectly traces out the dashed derivative curve in blue! Notice how every peak and valley on the orange curve lines up perfectly with a zero on the blue curve.

Until now, we have only looked at functions that depend on a single variable (like y depending on x). But in the real world, things often depend on two or more variables at once!

Take a cylinder. Its volume (V) depends on both its radius (r) and its height (h). The formula is V = πr²h.

If you want to know how fast the volume changes, you have to specify: changes when I do what? Do you mean when you stretch the cylinder taller? Or when you make it wider?

To solve this, Thompson introduces Partial Differentiation. The rule is incredibly simple: if you want to differentiate with respect to r, you just pretend h is a constant number (like 5), and differentiate normally! We write this as ∂V/∂r.

"A partial differential coefficient is simply one in which the differentiation has been performed with respect to one independent variable, while all the other independent variables have been kept constant."

Let's try it on V = πr²h:
1. Varying r (treat h as constant): ∂V/∂r = 2πrh.
2. Varying h (treat r as constant): ∂V/∂h = πr².

This isn't just abstract math - it has a beautiful physical meaning! Look at the interactive below. When you vary the radius, the added volume is a thin shell wrapped around the outside (Area = 2πrh). When you vary the height, the added volume is a thin flat slice added to the top (Area = πr²). The math perfectly matches the geometry!

Integration is differentiation's mirror. It means adding up an infinite number of infinitely thin slices to find a total - most famously, the area under a curve.

"The process of adding up the little bits of y · dx is called integration."

The symbol ∫ is simply a stretched-out S - standing for Sum. So ∫ y · dx reads literally as "the sum of all the little y · dx pieces." The notation tells you exactly what the operation does.

To find the exact area, divide the region into narrow rectangles. As they grow thinner and more numerous, the sum converges to the true value. The slider shows this convergence - watch imprecision settle into precision.

For y = x² from 0 to 2, the exact integral is 8/3 ≈ 2.667. Pull the slider to the right and watch the approximation approach it from above.

In the previous chapter, we learned that Integration is the process of adding up an infinite number of infinitely thin slices (finding the Area).

But integration has a second, equally important face: it is exactly the opposite of differentiation.

"If you differentiate y = x², you get dy/dx = 2x. Therefore, if you integrate 2x, you must get back to x²."

This mirror relationship is known as the Fundamental Theorem of Calculus. Differentiation and Integration are two halves of the exact same idea, like addition and subtraction, or multiplication and division.

Think about physics: if you know that a car's distance is x², you differentiate to find its speed is 2x. But if you only know its speed is 2x, you can integrate that speed over time to find the total distance traveled (x²)!

The interactive below proves this visually. It shows two graphs side-by-side. Move the slider to see how the Area of the top graph becomes the Height of the bottom graph. And simultaneously, the Height of the top graph is exactly the Slope of the bottom graph!

We now know that integrating is the reverse of differentiating. But how does this actually help us find the area under a curve?

Suppose we want to find the exact area under the curve y = x², starting from x = a and ending at x = b.

First, we integrate x² to find its "master" formula for area. By reversing the power rule, we know that integrating x² gives us x³/3.

"To find the area between two limits, you simply take the integral at the upper limit and subtract the integral at the lower limit."

This is called a Definite Integral. The area from a to b is simply the total area up to b, minus the "empty" area up to a.

So, the area is exactly (b³/3) - (a³/3).

Try it out in the interactive below. You can drag the lower bound (a) and the upper bound (b) to calculate the exact area under the curve y = x². Notice how the calculation flawlessly follows this simple subtraction rule!

Just as differentiation had its useful "dodges" (like the Chain Rule), integration has its own bag of tricks. But integration is notoriously trickier. While anyone can learn to differentiate almost anything by blindly following rules, integration requires a bit of art, intuition, and sometimes, a clever dodge.

Thompson covers several methods in this chapter, such as substitution and partial fractions. But the crowning triumph of integration dodges is Integration by Parts.

It stems from the Product Rule of differentiation. The formula looks like a strange algebra puzzle:

∫ u dv = uv - ∫ v du

It says: "If you have a really hard integral (∫ u dv), you can trade it for a completely different integral (∫ v du) minus a simple rectangle (uv). If you choose wisely, the new integral might be much easier to solve!"

This sounds like abstract magic, but it is actually a beautiful geometric truth. Imagine a curve plotted on a graph where the horizontal axis is u and the vertical axis is v.

In the interactive below, the Amber area represents the integral of v with respect to u. The Teal area represents the integral of u with respect to v. Together, they perfectly form a rectangle with area u × v. If you know the area of the rectangle, and you can calculate one of the shapes, you get the other one for free!

We have arrived at the grand finale of Thompson's book: Differential Equations.

Until now, we have mostly been given a function (like y = x²) and asked to find its slope (dy/dx), or vice versa. But what if we are given an equation that describes a relationship between a function and its slope, and we have to figure out what the function actually is?

For example: dy/dx = y.

This says: "Find me a function whose steepness is exactly equal to its height." As we learned in Chapter 14, the only function that does this is the organic growth curve, y = e^x.

Or how about dy/dx = -x/y? If you solve this, you get x² + y² = C, which is the equation for a circle!

"A differential equation is an equation containing differential coefficients. The process of finding what function it came from is called solving the differential equation."

But the most beautiful way to understand differential equations is to simply look at them. If an equation tells you the slope (dy/dx) at every point in space, you can draw a little line at every point showing that slope. This is called a Slope Field.

In the interactive below, choose a differential equation. The background shows its slope field. Click anywhere on the graph to drop an initial starting point. The computer will just follow the slopes, tracing out the hidden function right before your eyes!

In his famous Epilogue, Silvanus P. Thompson confidently predicts that professional mathematicians will damn his book as "thoroughly bad" for several reasons.

First and foremost, because it shows how ridiculously easy most calculus operations really are, giving away their closely guarded "trade secrets" and shattering the illusion of extreme difficulty.

To the critics who complain that he failed to use rigid, formal mathematical proofs, Thompson argues a simple point of pragmatism: "You don't forbid the use of a watch to every person who does not know how to make one." Just as children learn to speak before they learn rigid syntax, beginners should learn to use calculus before drowning in its proofs.

Finally, Thompson proudly pleads guilty to the accusation that he intentionally left out everything difficult. He wrote the book for "the legion of innocents" who had been deterred by the needlessly complicated way calculus is normally taught. His goal was to reveal its endearing simplicities and powerful methods without the irrelevant mathematical gymnastics.

"There are amongst young engineers a number on whose ears the adage that what one fool can do, another can, may fall with a familiar sound. They are earnestly requested not to give the author away, nor to tell the mathematicians what a fool he really is."