Derivatives and antiderivatives
Two questions, one single theory. How fast does a curve rise? That is the derivative. What area does it enclose? That is the integral. And the central theorem of the chapter says that one undoes what the other does.
0The symbols
A formula is not difficult. It is a sentence written in shorthand. Here is the dictionary.
| Symbol | Meaning |
|---|---|
| f(x) | The function: to each x, it assigns a value. The curve you draw. |
| f′(x) | "f prime": the derivative of f. It is a new function, giving the slope of the curve at every point. |
| Δx | "delta x": a gap between two x-values. The big Greek triangle always means "difference". |
| h | A small step. The same role as Δx, only shorter. It is meant to tend toward zero. |
| dy/dx | Another way of writing f′. It recalls where the derivative comes from: a quotient of two differences that have become infinitesimal. |
| lim | The value approached as h becomes infinitely small. You never set h = 0, you approach it. |
| F(x) | An antiderivative of f: a function whose derivative gives back f. The capital letter signals the reverse direction. |
| C | The constant of integration. Every antiderivative is defined up to a constant: the derivative of a constant is zero. |
| ∫ f(x) dx | "the integral of f": the set of antiderivatives of f. The elongated S is a summation S, we'll see why. |
| ∫ₐᵇ f(x) dx | The definite integral: a number. The area between the curve, the x-axis, and the two bounds a and b. |
| dx | The infinitesimal width of a slice. It also says with respect to which variable you are working. |
| e | Euler's number, about 2.718. The base for which the exponential function is its own derivative. |
| ln x | The natural logarithm: the inverse of eˣ. It answers "e to what power gives x?". |
1The derivative
A line has a slope: so much height gained per horizontal step. Always the same.
A curve does not have a slope. It has one at every point, and it keeps changing.
All of differentiation lies in this question: how do you measure the slope of a curve at a point?
1.1 · The difference quotient
We take two points of the curve: one at a, the other one step h further along.
We divide the rise by the run. That is the slope of the line joining these two points.
Because we only know how to measure slopes of straight lines.
So we cheat: we replace the curve with a line that crosses it. It is crude, but it is a start.
The smaller the step h, the more closely the line hugs the curve around a.
The approximation improves. It never becomes exact as long as h remains a real number.
With h = 0.1: 2.1. With h = 0.01: 2.01. It's heading toward 2.
1.2 · The definition of the derivative
We take the difference quotient. And we let the step tend to zero.
The limiting value is the derivative. The secant becomes the tangent.
You cannot set h = 0: that would mean dividing zero by zero.
The limit is the tool that gets around the obstacle: you watch where the quotient is heading, without ever actually landing there.
It's the same move as in the chapter on series. A limit rescuing an impossible calculation.
In practice, you almost never compute this limit by hand.
It has been worked out once and for all for every standard function. The result is the table in section 2.
The definition is there to help you understand the table, not to replace it.
1.3 · The equation of the tangent line
The line that touches the curve at a and has exactly its slope there.
It's a line: a point and a slope are all you need.
The point is (a, f(a)). The slope is f′(a). The formula says nothing more.
Three numbers to compute, in order: f(a), then f′(x), then f′(a).
Plug them into the formula. That's it.
y = 1 + 2(x − 1) = 2x − 1.
2The rules of differentiation
Differentiating requires no imagination at all. It's a construction kit.
A small table of ready-made derivatives, and four rules to assemble the pieces.
Every function in the syllabus can be differentiated with this.
2.1 · Standard derivatives
| f(x) | f′(x) | Worth noting |
|---|---|---|
| c | 0 | A constant doesn't rise: zero slope. |
| xⁿ | n · xⁿ⁻¹ | The exponent drops down front, then decreases by one. Also holds for negative or fractional n. |
| √x = x^½ | 1 / (2√x) | A special case of the line above, with n = ½. |
| 1/x = x⁻¹ | −1/x² | Same rule again, with n = −1. |
| eˣ | eˣ | Its own derivative. That is essentially the definition of the number e. |
| ln x | 1/x | The logarithm grows more and more slowly: its slope melts away like 1/x. |
| sin x | cos x | The cosine tells you how fast the sine is rising. |
| cos x | −sin x | Watch the minus sign. It's the most common slip in the chapter. |
2.2 · Sum and constant factor
You differentiate term by term. Multiplicative constants wait out front.
Slopes add up. If two effects overlap, so do their speeds.
And tripling a function triples its rise. Nothing deeper than that.
2.3 · Product and quotient
A product is not differentiated factor by factor.
Each factor gets differentiated in turn, while the other stays untouched, and you add the two contributions.
Think of a rectangle whose two sides are both growing.
Its area gains a strip along one side, and a strip along the other. Two contributions, one sum.
The quotient rule follows from this, with a minus sign and the square of the denominator.
Name the pieces before you calculate: f = …, g = …, then f′ and g′ separately.
Only then assemble them. Half the mistakes come from assembling in your head.
In the quotient, the order in the numerator matters: f′g first, minus fg′.
( x / (x+1) )′ = [ 1·(x+1) − x·1 ] / (x+1)² = 1 / (x+1)².
2.4 · The chain rule
A function inside a function: you differentiate the shell, leave the contents untouched, then multiply by the derivative of the contents.
Speeds multiply in cascade.
If g moves three times as fast as x, and f moves twice as fast as g, then f moves six times as fast as x.
Spot the shell and the contents. Say it out loud: "sine of something", "e to the power of something".
Differentiate the shell as if the contents were a plain letter. Then multiply by the derivative of the contents.
The reflex to build in: never a composite without its factor g′.
3Reading f through f′
The sign of the derivative tells the shape of the curve.
Positive: it's rising. Negative: it's falling. Zero: the curve is momentarily flat.
The derivative is the slope. A positive slope is a rise. By definition.
A maximum is a place where the rise stops and the fall begins: the derivative passes from plus to minus there, crossing zero.
To find the extrema: solve f′(x) = 0, then study the sign of f′ around each solution.
The sign table isn't a formality. It's what decides: maximum, minimum, or neither.
4Antiderivatives
It's differentiation in reverse. You know the slope everywhere, and you're looking for the curve.
Shift a curve upward: all its slopes stay the same.
So infinitely many curves share the same derivative. They differ only by a constant.
The + C says exactly that: the answer is a family, not a single curve.
You read the derivative table from right to left, adjusting the constants.
And you always check in the easy direction: differentiating the result should give back f. This check is free, do it every time.
| f(x) | ∫ f(x) dx | Check |
|---|---|---|
| xⁿ (n ≠ −1) | xⁿ⁺¹ / (n+1) + C | The exponent goes up by one, and then divides. |
| 1/x | ln |x| + C | The forbidden case n = −1 gets its own line. The absolute value covers negative x. |
| eˣ | eˣ + C | Unchanged, in both directions. |
| sin x | −cos x + C | The minus switches sides: (−cos)′ = sin. |
| cos x | sin x + C | No minus this time. |
Check: (x³ + 2x² − 5x + C)′ = 3x² + 4x − 5. ✓
5The definite integral
New question, seemingly unrelated: what is the area between a curve and the x-axis, from a to b?
The idea is the same as for series: cut into pieces you know how to measure, then refine endlessly.
5.1 · Area by strips
You cut the interval into n strips. On each one, you set up a rectangle that touches the curve.
You add up the areas of the rectangles. Then you refine: more strips, thinner ones.
We only know how to measure rectangles. Just as we only knew how to measure slopes of straight lines.
The detour is the same as in chapter 1, in reverse: approximate, then pass to the limit.
And that's why the sign ∫ is an S: it's the sum Σ, stretched out by the passage to the limit.
5.2 · The fundamental theorem
The area under the curve is computed without a single rectangle.
You take an antiderivative, evaluate it at the two bounds, and subtract. That's it.
Two seemingly unrelated problems, slope and area, turn out to be inverses of one another.
The area accumulated under f grows at exactly the speed f: differentiating the area gives back the function.
That's why the + C doesn't get in the way: in F(b) − F(a), the constant cancels itself out.
Three moves: find F, compute F(b), compute F(a). We write the whole thing [F(x)]ₐᵇ.
Watch the sign: area below the axis counts negatively. An integral can be negative or zero without the visible surface being zero.
The exact third that the lab above approximates with rectangles.
6The method
I've been asked to differentiate. What order to look in.
7The six mistakes
- Differentiating a product factor by factor. (f·g)′ is not f′·g′. Rectangle rule: two contributions, one sum.
- Forgetting the factor g′ in a composite. [sin(x²)]′ = cos(x²) · 2x, never just cos(x²).
- Concluding "extremum" as soon as f′(a) = 0. x³ has a zero derivative at 0 and no extremum. The sign around the point decides.
- Forgetting the + C. An antiderivative without + C is just one representative of the family. In a definite integral, however, no C.
- Integrating 1/x with the power rule. n = −1 is the forbidden case: dividing by n + 1 = 0. The answer is ln |x| + C.
- Mixing up the sign of cos′ and of ∫ sin. (cos x)′ = −sin x, and ∫ sin x dx = −cos x + C. The minus lives on both sides, but not in the same place.
8Test yourself
Six questions. Instant answer, with explanation.
9The vocabulary in French and German
This formulary comes from Swiss gymnases (upper secondary schools), where mathematics is taught in French or German depending on the canton. Here are the chapter's key terms in their original languages.
| Français | Deutsch | Remark |
|---|---|---|
| dériver | ableiten | The verb of the chapter. |
| la dérivée | die Ableitung | · |
| la pente | die Steigung | What the derivative measures. |
| le taux d'accroissement | der Differenzenquotient | The German name says what it is: a quotient of differences. |
| la tangente / la sécante | die Tangente / die Sekante | · |
| la règle du produit | die Produktregel | · |
| la règle du quotient | die Quotientenregel | · |
| la règle de la chaîne | die Kettenregel | The composite: shell times differentiated contents. |
| croissant / décroissant | steigend / fallend | · |
| un extremum, un maximum, un minimum | ein Extremum, ein Maximum, ein Minimum | · |
| la primitive | die Stammfunktion | Literally "stem function": the one that f descends from. |
| l'intégrale (définie) | das (bestimmte) Integral | · |
| la borne | die Grenze | a and b, the ends of the interval. |
| l'aire | der Flächeninhalt | · |
| la constante d'intégration | die Integrationskonstante | The famous + C. |
| le théorème fondamental | der Hauptsatz der Differential- und Integralrechnung | The bridge between the two halves of the chapter. |
See also the masterclass Sequences, sums and series: the limit, the tool shared by both chapters, is introduced there step by step.