Derivatives: Every Rule, With Worked Examples and When to Use Them

Derivatives almost never go wrong because of the formulas: they go wrong in deciding which formula to apply, and in what order. In our courses we see it every year in the fourth and fifth year: the student knows $D[\sin x] = \cos x$ by heart, then, faced with $\sin(3x^2)$, differentiates only the "outside" and loses the factor $6x$. This guide lines the whole thing up — the table of standard derivatives, the four rules, when to use each one, examples with the numbers — and closes with the mistakes we correct most often in lessons.

What are the derivatives of the standard functions?

Before the rules you need the building blocks: the derivatives of the standard functions. Here they are, and they have to be known without hesitation, because every exercise uses them as ingredients.

A detail we repeat often in lessons: $\sqrt{x}$ and $\frac{1}{x}$ are not special cases to be learned separately. They are the power rule applied with $n = \frac{1}{2}$ and $n = -1$. Anyone who sees them that way also gets $\frac{1}{x^2} \to -\frac{2}{x^3}$ or $\sqrt[3]{x} \to \frac{1}{3}x^{-2/3}$ on the spot, with no extra formula sheets.

And if you are wondering where these formulas come from: $D[\sin x] = \cos x$ is proved using precisely the standard limit $\frac{\sin x}{x} \to 1$ — if you want to revisit it, we cover it in our guide to standard limits for the Maturità.

How do the four rules of differentiation work?

Sum and difference. The derivative of a sum is the sum of the derivatives:

and multiplicative constants come outside the derivative: $D[k \cdot f(x)] = k \cdot f'(x)$. Together, these two properties let you differentiate any polynomial term by term.

Product (Leibniz's rule). When two functions that both contain $x$ are multiplied together:

"Differentiate the first times the second untouched, plus the first untouched times the derivative of the second." Here the order makes no difference: the sum is symmetric.

Quotient. For a fraction with $x$ both above and below:

Here, by contrast, the order in the numerator matters a great deal: first the derivative of the numerator times the denominator, then the minus. Swapping the two terms flips the sign of the entire answer.

Composite function (chain rule). When one function sits "inside" another, that is $f(g(x))$:

In practice: you differentiate the outer function leaving the argument untouched, then multiply by the derivative of the argument. That second factor — the derivative of the "inside" — is the piece students forget more than anything else, and we come back to it shortly.

Which rule should I use? The decision table

The real work is not remembering the formulas but classifying the function at a glance. The most reliable test is to read the function out loud: if you say "sine of...", "e to the power of...", "square root of..." and what follows is not just $x$ on its own, you have a composite function.

Four worked examples, with the numbers

Polynomial with a root and a negative power

No product, no composite: we rewrite everything as powers, $f(x) = 3x^4 - 2x^{1/2} + 5x^{-1}$, and differentiate term by term:

Quick check at a point: $f'(1) = 12 - 1 - 5 = 6$. Getting into the habit of checking the derivative at a convenient value is a lifesaver in class tests.

The quotient rule

$x$ appears above and below: quotient. Numerator: $f' = 2x$; denominator: $g' = 2$.

Had we swapped the order in the numerator we would have got $\frac{-2x^2+6x+2}{(2x-3)^2}$: the same absolute value, the opposite sign everywhere. In curve sketching that means swapping the increasing and decreasing intervals — the whole graph comes out wrong because of one reversed order.

The chain rule

Let us read it out loud: "sine of $3x^2$" — composite. Outside: the sine, which differentiates to the cosine of the same argument. Inside: $3x^2$, which differentiates to $6x$.

Writing only $\cos(3x^2)$ is the single most frequent mistake we see on this topic: the derivative of the "inside" is not optional.

Product and chain together — the case that catches almost everyone

Two factors containing $x$: the product rule is needed. But the second factor is itself composite ($\sin$ of $3x$), so differentiating it also calls for the chain rule:

The two classic traps here: forgetting the factor $3$ inside the product, or "merging" the rules and writing $2x \cdot 3\cos(3x)$ straight off, differentiating both factors together. The rules are applied one at a time, from the outside in.

Do you always need the quotient rule?

No, and using it when you do not need it is a source of avoidable mistakes. If there is a constant on top, the fraction is a power in disguise:

One line, against the four of the quotient rule (with the attendant risk of getting a sign wrong). Same thing for $\frac{1}{\sqrt{x}} = x^{-1/2}$ or $\frac{3}{2x} = \frac{3}{2}x^{-1}$. The practical rule we give our students: use the quotient rule only when $x$ really is both above and below; in every other case, rewrite as a power and reach for a standard derivative.

The mistakes we see most often in lessons

Marking fourth-year tests and Maturità mock papers, five mistakes come back with more regularity than any others:

1. The chain multiplier forgotten. $D[e^{x^2}]$ written as $e^{x^2}$ rather than $2x\,e^{x^2}$; $D[\sin 3x]$ written as $\cos 3x$ rather than $3\cos 3x$. It is the number one mistake overall, and the most insidious: the answer looks like a legitimate derivative, so nobody catches it on rereading.
2. The order reversed in the quotient rule. The numerator is $f'g - fg'$, in that order. Reversing it produces the answer with the opposite sign — and in curve sketching the whole increasing/decreasing analysis comes out backwards, as in the example of $\frac{x^2+1}{2x-3}$ above.
3. The product differentiated "factor by factor". $D[x^2 \ln x] = 2x \cdot \frac{1}{x} = 2$: elegant, tidy, wrong. Leibniz's rule gives $2x\ln x + x$. The fact that the wrong answer comes out so neatly makes it even more treacherous.
4. Constants treated as variables. $D[\ln 3] = \frac{1}{3}$ and $D[e^2] = 2e$ are mistakes we still see in the fifth year: $\ln 3$, $e^2$ and $\pi^2$ are numbers, and the derivative of a number is $0$.
5. The sign lost with negative powers. $D\!\left[\frac{5}{x}\right] = \frac{5}{x^2}$ without the minus: whoever writes that has applied $n\,x^{n-1}$ while forgetting that here $n = -1$, which makes the coefficient negative. The correct answer is $-\frac{5}{x^2}$.

From derivatives to the Maturità (and beyond)

Derivatives are the heart of curve sketching, which in turn is the centrepiece of the seconda prova, the Maturità maths paper at a scientific high school: maxima, minima, points of inflection and monotonicity all go through an $f'$ calculated correctly. And it does not end with the exam: the same rules, unchanged, are waiting for you in the calculus questions of admission tests and in your first university analysis exam — anyone who automates them now arrives in first year with a real advantage. If you are already thinking about admission tests, take a look at our preparation paths.

To fix them for good, the same recipe as for limits works: a few minutes of rapid-fire derivatives at the start of a lesson, mixing the four types, until the S/P/Q/C classification becomes instinctive. If instead you feel the gaps go further back — the algebra of powers, functions, limits — a dedicated tutor can build you a plan around the weeks you have ahead: tell us about your situation and book your first lesson.

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