--- title: What is a Derivative? - Calculus Basics Explained Simply description: "Learn what a derivative is in calculus: its definition as a rate of change, its connection to the slope of a tangent, and basic properties." url: https://astra-ai.co/videos/derivatives/definition-of-the-derivative/what-is-a-derivative/ locale: en type: video-lesson published: 2026-10-01T09:59:49.867Z updated: 2026-10-01T09:59:49.867Z alternates: bg: https://astra-ai.co/bg/videos/derivati/definitsiya-na-derivatsiya/kakvo-e-derivatsiya/ br: https://astra-ai.co/br/videos/derivadas/capitulo-matematica-050/video-matematica-099/ cs: https://astra-ai.co/cs/videos/derivaty/definice-derivace/co-je-derivace/ da: https://astra-ai.co/da/videos/differentialregning/definition-af-differentialkvotienten/hvad-er-en-afledt/ de: https://astra-ai.co/de/videos/ableitungen/definition-der-ableitung/was-ist-eine-ableitung/ el: https://astra-ai.co/el/videos/paragogoi/orismos-tis-paragogou/what-is-a-derivative/ en: https://astra-ai.co/videos/derivatives/definition-of-the-derivative/what-is-a-derivative/ en-IN: https://astra-ai.co/en-IN/videos/derivatives/chapter-d3fc5af6/what-is-a-derivative/ en-PH: https://astra-ai.co/en-PH/videos/derivatives/chapter-d3fc5af6/lesson-8fb0e77c/ es: https://astra-ai.co/es/videos/derivadas/definicion-de-la-derivada/que-es-una-derivada/ es-AR: https://astra-ai.co/es-AR/videos/derivadas/capitulo-44/leccion-de-matematicas-99/ es-CL: https://astra-ai.co/es-CL/videos/derivadas/capitulo-44/leccion-de-matematicas-99/ es-PE: https://astra-ai.co/es-PE/videos/derivadas/capitulo-video-d3fc5af6/leccion-de-video-8fb0e77c/ et: https://astra-ai.co/et/videos/tuletised/tuletise-definitsioon/mis-on-tuletis/ fi: https://astra-ai.co/fi/videos/derivaatat/derivaatan-maaritelma/mika-on-derivaatta/ fr: https://astra-ai.co/fr/videos/la-derivee/definition-de-la-derivee/quest-ce-quune-derivee/ hr: https://astra-ai.co/hr/videos/derivacije/definicija-derivacije/sto-je-derivacija/ hu: https://astra-ai.co/hu/videos/derivaltak/a-derivalt-definicioja/mi-a-derivalt/ id: https://astra-ai.co/id/videos/turunan/definisi-turunan/apa-itu-turunan/ it: https://astra-ai.co/it/videos/derivate/definizione-della-derivata/cose-una-derivata/ ja: https://astra-ai.co/ja/videos/derivatives-ja/44/099/ km: https://astra-ai.co/km/videos/derivatives/chapter-d3fc5af6/lesson-8fb0e77c/ ko: https://astra-ai.co/ko/videos/derivatives-ko/44/099/ lo: https://astra-ai.co/lo/videos/lo-d351a5ea/lo-d3fc5af6/lesson-8fb0e77c/ lt: https://astra-ai.co/lt/videos/isvestines/isvestines-apibrezimas/pamoka-8fb0e77c/ lv: https://astra-ai.co/lv/videos/atvasinajumi/atvasinajuma-definicija/stunda-8fb0e77c/ ms: https://astra-ai.co/ms/videos/terbitan/bab-50-matematik-asas/pelajaran-099/ ms-BN: https://astra-ai.co/ms-BN/videos/terbitan/bab-d3fc5af6/pelajaran-8fb0e77c/ mx: https://astra-ai.co/mx/videos/derivadas/capitulo-44/leccion-99/ my: https://astra-ai.co/my/videos/my-d351a5ea/my-d3fc5af6/lesson-8fb0e77c/ nl: https://astra-ai.co/nl/videos/derivaten/definitie-van-de-afgeleide/wat-is-een-afgeleide/ no: https://astra-ai.co/no/videos/deriverte/definisjonen-av-den-deriverte/hva-er-en-derivert/ pl: https://astra-ai.co/pl/videos/pochodne/definicja-pochodnej/czym-jest-pochodna/ pt: https://astra-ai.co/pt/videos/derivadas/definicao-de-derivada/o-que-e-derivacao/ pt-TL: https://astra-ai.co/pt-TL/videos/derivadas/capitulo-de-video-044/licao-de-video-099/ ro: https://astra-ai.co/ro/videos/derivate/definitia-derivatei/ce-este-o-derivata/ sk: https://astra-ai.co/sk/videos/derivacie/kapitola-044/lekcia-099/ sl: https://astra-ai.co/sl/videos/odvodi/definicija-odvoda/kaj-je-odvod/ sr: https://astra-ai.co/sr/videos/izvodi/poglavlje-044/lekcija-099/ sv: https://astra-ai.co/sv/videos/derivator/definition-av-derivatan/vad-ar-en-derivata/ th: https://astra-ai.co/th/videos/anuphan/bot-50-niyam-anuphan/botrian-099/ tr: https://astra-ai.co/tr/videos/turevler/bolum-044/ders-099/ uk: https://astra-ai.co/uk/videos/uk-topic-13/uk-chapter-50/uk-lesson-8fb0e77c/ vi: https://astra-ai.co/vi/videos/vi-topic-13/vi-chapter-50/ao-ham-la-gi/ zh-HK: https://astra-ai.co/zh-HK/videos/shuxue-kecheng-013/shuxue-zhang-050/shuxue-kecheng-099/ zh-TW: https://astra-ai.co/zh-TW/videos/derivatives/chapter-d3fc5af6/lesson-8fb0e77c/ --- # What is a Derivative? Derivatives are a fundamental concept in mathematics, particularly in differential calculus, that describes how the value of a function changes as its input values change. In other words, the derivative of a function at a given point measures the slope of the tangent line to the graph of the function at that point and is central to understanding the dynamics of change. ## DEFINITION Mathematically, the derivative of a function 'f' with respect to a variable 'x' is expressed as the limit of the difference quotient of the function as the difference 'h' between two values of 'x' approaches zero. (f'(x) = lim (as h→0) \[f(x+h) - f(x)]/h). ## PROPERTIES OF DERIVATIVES - **LINEARITY:** The derivative of a sum of functions is equal to the sum of the derivatives of the functions. ( (u+v)' = u' + v' ) - **PRODUCT RULE:** The derivative of the product of two functions is determined by a rule that includes the derivatives of both functions. ( (uv)' = u'v + uv' ) - **CHAIN RULE:** Allows for the calculation of the derivative of composite functions. ( (f(g(x)))' = f'(g(x)) \* g'(x) ) - **DERIVATIVE OF AN INVERSE FUNCTION:** Defines the relationship between the derivative of a function and its inverse. ## SLOPE OF THE TANGENT AND THE DERIVATIVE The slope of the tangent to the graph of a function at point 'x' is actually the derivative of the function at that point. If we draw a tangent line to the graph of the function at point 'x', the slope (or steepness) of this tangent will be equal to the value of the derivative f'(x) at point 'x'. ### HOW TO UNDERSTAND THE CONNECTION? Imagine you are graphing a function, for example, f(x) = x². When you draw a tangent line to the graph of this function at a specific point, for instance, at x = 3, this line will have a certain slope. This slope of the line is the value of the derivative f'(x) at point x = 3. For our example, f'(x) = 2x, which means that f'(3) = 6. This tells us that the slope of the tangent to the graph of f(x) = x² at the point x = 3 is 6. ## EXAMPLE OF USE: Let's take the function f(x) = x². Its derivative is f'(x) = 2x. If we want to find the slope of the tangent at the point x = 2, we simply calculate the derivative at that point: f'(2) = 2 \* 2 = 4. Therefore, the slope of the tangent to the graph of the function f(x) = x² at the point x = 2 is 4.