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Updated Apr 4, 2026
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Nombre Dérivé et Applications Pratiques
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Le nombre dérivé - Introduction
Le nombre dérivé est partout autour de toi ! Imagine que tu regardes la courbe d'une fonction sur ton écran - le nombre dérivé te dit exactement à quelle vitesse cette courbe monte ou descend à n'importe quel point.
Pour le calculer, on utilise d'abord le taux d'accroissement. C'est la formule magique : /h. Cette formule compare la fonction entre deux points très proches.
💡 Astuce pratique : Le taux d'accroissement, c'est comme calculer la vitesse moyenne entre deux feux rouges - sauf qu'ici on s'intéresse aux fonctions !
Exercice 1 - Fonction quadratique
Avec la fonction f(x) = x² - 3x + 1, tu vas voir que les calculs deviennent plus simples qu'ils n'y paraissent ! D'abord, on trouve le taux d'accroissement : 2a + h - 3.
Ensuite, pour obtenir le nombre dérivé f'(a), on fait tendre h vers 0. Résultat : f'(a) = 2a - 3. C'est tout !
Pour l'équation de la tangente au point x = 2, on applique la formule y = f'(a) + f(a). Avec nos calculs, ça donne y = x - 3.
💡 Rappel important : La tangente "touche" la courbe en un seul point et a la même pente que la courbe à cet endroit !
Exercice 2 - Interprétation graphique
Quand on te donne une tangente avec son équation y = -2x + 5 au point A(1;3), tu peux en tirer plein d'infos ! Le point A nous dit directement que g(1) = 3.
Le coefficient directeur -2 de la tangente, c'est exactement la valeur de g'(1). Donc g'(1) = -2. Simple comme bonjour !
Puisque g'(1) = -2 < 0, la fonction est décroissante au voisinage du point A. Un nombre dérivé négatif = fonction qui descend !
💡 Truc de mémorisation : Coefficient directeur positif → fonction qui monte, négatif → fonction qui descend !
Exercice 3 - Application physique
Avec un projectile de hauteur h(t) = -5t² + 20t + 1, tu découvres que la vitesse instantanée est la dérivée de la hauteur ! Donc v(t) = -10t + 20.
À t = 1 seconde, la vitesse est v(1) = 10 m/s. Le projectile monte encore à ce moment-là !
Pour trouver la hauteur maximale, on cherche quand v(t) = 0. Ça donne -10t + 20 = 0, donc t = 2 secondes. À ce moment précis, le projectile s'arrête de monter avant de redescendre.
💡 Astuce physique : En physique, la dérivée de la position donne la vitesse. Quand la vitesse = 0, on est à un extremum !
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What is the Knowunity AI companion?
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
Where can I download the Knowunity app?
You can download the app from Google Play Store and Apple App Store.
Is Knowunity really free of charge?
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
Most popular content: test de la première dérivée
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The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
Nombre Dérivé et Applications Pratiques
Tu vas découvrir le nombre dérivé, un concept super utile pour comprendre comment les fonctions changent ! C'est comme mesurer la vitesse d'une voiture à un instant précis ou trouver la pente d'une montagne à un endroit donné.
Le nombre dérivé - Introduction
Le nombre dérivé est partout autour de toi ! Imagine que tu regardes la courbe d'une fonction sur ton écran - le nombre dérivé te dit exactement à quelle vitesse cette courbe monte ou descend à n'importe quel point.
Pour le calculer, on utilise d'abord le taux d'accroissement. C'est la formule magique : /h. Cette formule compare la fonction entre deux points très proches.
💡 Astuce pratique : Le taux d'accroissement, c'est comme calculer la vitesse moyenne entre deux feux rouges - sauf qu'ici on s'intéresse aux fonctions !
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Exercice 1 - Fonction quadratique
Avec la fonction f(x) = x² - 3x + 1, tu vas voir que les calculs deviennent plus simples qu'ils n'y paraissent ! D'abord, on trouve le taux d'accroissement : 2a + h - 3.
Ensuite, pour obtenir le nombre dérivé f'(a), on fait tendre h vers 0. Résultat : f'(a) = 2a - 3. C'est tout !
Pour l'équation de la tangente au point x = 2, on applique la formule y = f'(a) + f(a). Avec nos calculs, ça donne y = x - 3.
💡 Rappel important : La tangente "touche" la courbe en un seul point et a la même pente que la courbe à cet endroit !
Sign up to see the contentIt's free!
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Improve your grades
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Exercice 2 - Interprétation graphique
Quand on te donne une tangente avec son équation y = -2x + 5 au point A(1;3), tu peux en tirer plein d'infos ! Le point A nous dit directement que g(1) = 3.
Le coefficient directeur -2 de la tangente, c'est exactement la valeur de g'(1). Donc g'(1) = -2. Simple comme bonjour !
Puisque g'(1) = -2 < 0, la fonction est décroissante au voisinage du point A. Un nombre dérivé négatif = fonction qui descend !
💡 Truc de mémorisation : Coefficient directeur positif → fonction qui monte, négatif → fonction qui descend !
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Exercice 3 - Application physique
Avec un projectile de hauteur h(t) = -5t² + 20t + 1, tu découvres que la vitesse instantanée est la dérivée de la hauteur ! Donc v(t) = -10t + 20.
À t = 1 seconde, la vitesse est v(1) = 10 m/s. Le projectile monte encore à ce moment-là !
Pour trouver la hauteur maximale, on cherche quand v(t) = 0. Ça donne -10t + 20 = 0, donc t = 2 secondes. À ce moment précis, le projectile s'arrête de monter avant de redescendre.
💡 Astuce physique : En physique, la dérivée de la position donne la vitesse. Quand la vitesse = 0, on est à un extremum !
Sign up to see the contentIt's free!
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Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
Where can I download the Knowunity app?
You can download the app from Google Play Store and Apple App Store.
Is Knowunity really free of charge?
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
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Most popular content: test de la première dérivée
Most popular content in Maths
Fiches récapitulatives spé maths - TOUT le programme de terminale
Ces fiches vont vous sauver pour le bac de spé maths! :)
MathsTle
Most popular content
Can't find what you're looking for? Explore other subjects.
Students love us — and so will you.
4.6/5
App Store
4.7/5
Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user