--- title: Derivatives - Understanding Rate of Change (Exercise 3) description: "A derivative exercise, the third set: practice the definition and the basic rules on new functions." url: https://astra-ai.co/videos/derivatives/definition-of-the-derivative/derivative-exercise-3/ locale: en type: video-lesson keywords: - derivative - derivative exercise - differentiation practice - worked example published: 2026-10-01T10:03:21.317Z updated: 2026-10-04T22:01:22.595Z alternates: bg: https://astra-ai.co/bg/videos/derivati/definitsiya-na-derivatsiya/izvezhdane-uprazhnenie-3/ br: https://astra-ai.co/br/videos/derivadas/capitulo-matematica-050/video-matematica-029/ cs: https://astra-ai.co/cs/videos/derivaty/definice-derivace/derivace-cviceni-3/ da: https://astra-ai.co/da/videos/differentialregning/definition-af-differentialkvotienten/differentialkvotient-velse-3/ de: https://astra-ai.co/de/videos/ableitungen/definition-der-ableitung/ableitung-aufgabe-3/ el: https://astra-ai.co/el/videos/paragogoi/orismos-tis-paragogou/derivative-exercise-3/ en: https://astra-ai.co/videos/derivatives/definition-of-the-derivative/derivative-exercise-3/ en-IN: https://astra-ai.co/en-IN/videos/derivatives/chapter-d3fc5af6/derivative-exercise-3/ en-PH: https://astra-ai.co/en-PH/videos/derivatives/chapter-d3fc5af6/lesson-ada5b461/ es: https://astra-ai.co/es/videos/derivadas/definicion-de-la-derivada/derivadas-ejercicio-3/ es-AR: https://astra-ai.co/es-AR/videos/derivadas/capitulo-44/leccion-29/ es-CL: https://astra-ai.co/es-CL/videos/derivadas/capitulo-44/leccion-29/ es-PE: https://astra-ai.co/es-PE/videos/derivadas/capitulo-video-d3fc5af6/leccion-de-video-ada5b461/ et: https://astra-ai.co/et/videos/tuletised/tuletise-definitsioon/tuletis-harjutus-3/ fi: https://astra-ai.co/fi/videos/derivaatat/derivaatan-maaritelma/derivaatta-harjoitus-3/ fr: https://astra-ai.co/fr/videos/la-derivee/definition-de-la-derivee/derive-exercice-3/ hr: https://astra-ai.co/hr/videos/derivacije/definicija-derivacije/derivacija-vjezba-3/ hu: https://astra-ai.co/hu/videos/derivaltak/a-derivalt-definicioja/derivalas-3-gyakorlat/ id: https://astra-ai.co/id/videos/turunan/definisi-turunan/turunan-latihan-3/ it: https://astra-ai.co/it/videos/derivate/definizione-della-derivata/derivata-esercizio-3/ ja: https://astra-ai.co/ja/videos/derivatives-ja/44/029/ km: https://astra-ai.co/km/videos/derivatives/chapter-d3fc5af6/lesson-ada5b461/ ko: https://astra-ai.co/ko/videos/derivatives-ko/44/029/ lo: https://astra-ai.co/lo/videos/lo-d351a5ea/lo-d3fc5af6/lo-ada5b461/ lt: https://astra-ai.co/lt/videos/isvestines/isvestines-apibrezimas/pamoka-ada5b461/ lv: https://astra-ai.co/lv/videos/atvasinajumi/atvasinajuma-definicija/stunda-ada5b461/ ms: https://astra-ai.co/ms/videos/terbitan/bab-50-matematik-asas/pelajaran-029/ ms-BN: https://astra-ai.co/ms-BN/videos/terbitan/bab-d3fc5af6/pelajaran-ada5b461/ mx: https://astra-ai.co/mx/videos/derivadas/capitulo-44/leccion-29/ my: https://astra-ai.co/my/videos/my-d351a5ea/my-d3fc5af6/my-ada5b461/ nl: https://astra-ai.co/nl/videos/derivaten/definitie-van-de-afgeleide/afgeleide-oefening-3/ no: https://astra-ai.co/no/videos/deriverte/definisjonen-av-den-deriverte/derivasjon-velse-3/ pl: https://astra-ai.co/pl/videos/pochodne/definicja-pochodnej/pochodna-cwiczenie-3/ pt: https://astra-ai.co/pt/videos/derivadas/definicao-de-derivada/derivacao-exercicio-3/ pt-TL: https://astra-ai.co/pt-TL/videos/derivadas/capitulo-de-video-044/licao-de-video-029/ ro: https://astra-ai.co/ro/videos/derivate/definitia-derivatei/lectie-video-ada5b461/ sk: https://astra-ai.co/sk/videos/derivacie/kapitola-044/lekcia-029/ sl: https://astra-ai.co/sl/videos/odvodi/definicija-odvoda/odvod-vaja-3/ sr: https://astra-ai.co/sr/videos/izvodi/poglavlje-044/lekcija-029/ sv: https://astra-ai.co/sv/videos/derivator/definition-av-derivatan/derivata-ovning-3/ th: https://astra-ai.co/th/videos/anuphan/bot-50-niyam-anuphan/botrian-029/ tr: https://astra-ai.co/tr/videos/turevler/bolum-044/ders-029/ uk: https://astra-ai.co/uk/videos/uk-topic-13/uk-chapter-50/uk-lesson-ada5b461/ vi: https://astra-ai.co/vi/videos/vi-topic-13/vi-chapter-50/ao-ham-bai-tap-3/ zh-HK: https://astra-ai.co/zh-HK/videos/shuxue-kecheng-013/shuxue-zhang-050/shuxue-kecheng-029/ zh-TW: https://astra-ai.co/zh-TW/videos/derivatives/chapter-d3fc5af6/lesson-ada5b461/ --- # Derivative – Exercise 3 > This derivative exercise is a third set of practice. It applies the definition and the basic rules to new functions. Calculating the derivatives of functions is a key part of differential calculus in mathematics, which focuses on analyzing how the values of functions change with respect to changes in their variables. The derivative of a function tells us the rate of change of the function's value when the independent variable changes slightly. This mathematical process is a fundamental building block for understanding and modeling dynamic systems. ## BASICS OF DERIVATIVES The derivative of a function with respect to a variable is expressed as f′(x) or df/dx, which represents the limit of the ratio between an infinitesimally small change in the function and an infinitesimally small change in the variable. This concept of a limit is the basis for calculating derivatives and allows for precise measurement of how a function responds to changes. ## HOW DO WE CALCULATE DERIVATIVES? When calculating derivatives, we use basic procedures that follow the definition of a limit. These procedures allow us to derive a new equation from the function's equation, which describes its derivative. It is crucial to understand that the derivative of a function at a specific point indicates how steeply the function is changing at that point. ## IMPORTANCE OF CALCULATING DERIVATIVES Knowledge of derivatives enables the solving of practical problems, such as determining velocity or acceleration in physics, finding points where a function reaches its maximum or minimum, and analyzing the changes in various physical and economic systems. Derivatives are also key in finding the tangent to a curve at a specific point, which plays an important role in geometry and computer graphics. Understanding and the ability to calculate derivatives open the door to comprehending complex mathematical concepts.