Free · Year 5 of high school · 50 minutes · 15 questions · no penalty

Practice maths class test — Year 5: derivatives.

15 questions drawn at random from a bank of 25, on the block of the programme your teacher is finishing: every attempt is a different test. No penalty for wrong answers, so answer even if you are not sure. At the end you receive by email the correct answer to every question and your weak topics — with an account you see them straight away here on the page. The step-by-step solution unlocks with your first lesson.

A class test is the written test an Italian high-school class sits after each block of the programme; this practice version is multiple choice.

Want to see the reasoning before you try? Look atthe secant that becomes a tangent, in an interactive figure.

Test in ItalianThe questions of this simulator are in Italian, as in the real exam.

Test prepared by Gaetano Livornese · Aerospace engineer, Politecnico di Milano · maths and physics tutor · updated 13 October 2026

Programme covered by the test

The difference quotient and the definition of the derivative
Basic derivatives and differentiation rules
Derivative of a composite function and of an inverse
The tangent line and its geometric meaning
Points where the function is not differentiable: corners, cusps, vertical inflection points
Rolle’s and Lagrange’s theorems

Two worked examples from the test

Question 1 · Derivate: regole, composta, non derivabilità, teoremi · Derivate e regole di derivazione

Per la funzione $f (x) = x^{2}$ il rapporto incrementale relativo al punto $x_{0} = 3$ e all'incremento $h$ (con $h \neq = 0$ ) vale

A $6 h + h^{2}$
B $6 + h$ correct answer
C $6$
D $9 + h$

Solution

Calcolo $f (3 + h) = (3 + h)^{2} = 9 + 6 h + h^{2}$ e $f (3) = 9$ .
Il rapporto incrementale è $\frac{f ( 3 + h ) - f ( 3 )}{h} = \frac{6 h + h ^{2}}{h} .$
Raccolgo $h$ a numeratore: $\frac{h ( 6 + h )}{h}$ .
Semplifico (lecito perché $h \neq = 0$ ): resta $6 + h$ .
Attenzione: $6 h + h^{2}$ è solo il numeratore, la divisione per $h$ non è stata fatta. Il valore $6$ si ottiene solo dopo il limite per $h \to 0$ , cioè è $f^{'} (3)$ , non il rapporto incrementale.

Question 2 · Derivate: regole, composta, non derivabilità, teoremi · Derivate e regole di derivazione

Il limite

$h \to 0 lim \frac{4 + h - 2}{h}$

è, per definizione, la derivata di una funzione elementare in un punto. Quale funzione, in quale punto, e quanto vale il limite?

A $f (x) = x$ in $x_{0} = 4$ ; il limite vale $\frac{1}{4}$ correct answer
B $f (x) = x$ in $x_{0} = 4$ ; il limite vale $\frac{1}{2}$
C $f (x) = x$ in $x_{0} = 2$ ; il limite vale $\frac{1}{2 2}$
D $f (x) = x^{2}$ in $x_{0} = 2$ ; il limite vale $4$

Solution

Confronto con la definizione $f^{'} (x_{0}) = h \to 0 lim \frac{f ( x _{0} + h ) - f ( x _{0} )}{h} .$
Al numeratore compare $4 + h$ , cioè $f (x_{0} + h)$ con $f (x) = x$ e $x_{0} = 4$ .
Coerentemente $f (x_{0}) = 4 = 2$ , che è proprio il numero sottratto.
Quindi il limite vale $f^{'} (4) = \frac{1}{2 4} = \frac{1}{4} .$
Il $2$ che si legge nel testo è il valore della funzione, non il punto: prendere $x_{0} = 2$ è l'errore più frequente.

In the online practice test you will find 13 more questions like these; the correct answer to every question arrives by email.

Prefer paper? Download the test as a PDF (15 questions, 4 pages · 80 KB, ready to print). You receive the solutions by email after taking it online.

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What a maths test on derivatives looks like

Derivatives with the product, quotient and chain rules, at least one derivative to compute from the definition, the tangent line at a point, the classification of the points where the function is not differentiable and an application of Rolle’s or Lagrange’s theorem with the hypotheses to check.

The mistake that costs the grade

Forgetting the inner derivative in a composite function. On theorems, applying Rolle or Lagrange without checking the hypotheses: continuity on the closed interval and differentiability on the open one must be verified explicitly, and that is half the marks of the exercise.

How to use this practice test

Take it once cold, without revising and with the timer on: it is there to show you where you stand, not to look good. Then look at your weak topics, revise only those — usually two or three, not the whole chapter — and retake it after a few days. The 15 questions are drawn from a bank of 25, so the second attempt is not the same test and the scores are genuinely comparable.

Is the score not enough?

See which topics lost you the most marks, then read the guide to derivatives explained well — or we look at your results together: the diagnostic is free.

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