--- title: Derivatives of Elementary Functions - Formulas & Rules description: "Derivatives of elementary functions: exponentials, logs, and trig functions, using the standard rules." url: https://astra-ai.co/videos/derivatives/derivatives-of-other-elementary-functions/ locale: en type: video-chapter keywords: - derivatives of elementary functions - derivative rules - exponential derivative - trigonometric derivatives published: 2026-10-01T09:47:38.338Z updated: 2026-10-04T21:59:21.327Z alternates: it: https://astra-ai.co/it/videos/derivate/derivate-altre-funzioni-elementari/ de: https://astra-ai.co/de/videos/ableitungen/ableitungen-anderer-elementarer-funktionen/ es: https://astra-ai.co/es/videos/derivadas/derivadas-de-funciones-elementales/ fr: https://astra-ai.co/fr/videos/la-derivee/derivees-des-fonctions-elementaires/ pl: https://astra-ai.co/pl/videos/pochodne/pochodne-innych-funkcji-elementarnych/ sl: https://astra-ai.co/sl/videos/odvodi/odvodi-elementarnih-funkcij/ fi: https://astra-ai.co/fi/videos/derivaatat/muiden-alkeisfunktioiden-derivaatat/ hr: https://astra-ai.co/hr/videos/derivacije/derivacije-elementarnih-funkcija/ el: https://astra-ai.co/el/videos/paragogoi/paragogoi-allon-stoicheiodon-synartiseon/ pt: https://astra-ai.co/pt/videos/derivadas/derivadas-de-funcoes-elementares/ nl: https://astra-ai.co/nl/videos/derivaten/afgeleiden-van-andere-elementaire-functies/ da: https://astra-ai.co/da/videos/differentialregning/afledte-af-andre-elementaere-funktioner/ bg: https://astra-ai.co/bg/videos/derivati/proizvodni-na-elementarni-funktsii/ cs: https://astra-ai.co/cs/videos/derivaty/derivace-elementarnich-funkci/ ja: https://astra-ai.co/ja/videos/derivatives-ja/31/ id: https://astra-ai.co/id/videos/turunan/turunan-fungsi-elementer/ et: https://astra-ai.co/et/videos/tuletised/teiste-elementaarfunktsioonide-tuletised/ hu: https://astra-ai.co/hu/videos/derivaltak/elemi-fuggvenyek-derivaltjai/ uk: https://astra-ai.co/uk/videos/uk-topic-13/uk-chapter-37/ sk: https://astra-ai.co/sk/videos/derivacie/kapitola-031/ vi: https://astra-ai.co/vi/videos/vi-topic-13/vi-chapter-37/ my: https://astra-ai.co/my/videos/my-d351a5ea/my-311c891c/ es-CL: https://astra-ai.co/es-CL/videos/derivadas/capitulo-31/ ko: https://astra-ai.co/ko/videos/derivatives-ko/31/ sv: https://astra-ai.co/sv/videos/derivator/derivator-av-andra-elementara-funktioner/ ro: https://astra-ai.co/ro/videos/derivate/derivatele-altor-functii-elementare/ no: https://astra-ai.co/no/videos/deriverte/deriverte-av-andre-element-re-funksjoner/ sr: https://astra-ai.co/sr/videos/izvodi/poglavlje-031/ lt: https://astra-ai.co/lt/videos/isvestines/kitu-elementariuju-funkciju-isvestines/ lv: https://astra-ai.co/lv/videos/atvasinajumi/citu-elementaro-funkciju-atvasinajumi/ tr: https://astra-ai.co/tr/videos/turevler/bolum-031/ zh-TW: https://astra-ai.co/zh-TW/videos/derivatives/chapter-311c891c/ th: https://astra-ai.co/th/videos/anuphan/bot-37-anuphan-fangchan-phuenthan/ ms: https://astra-ai.co/ms/videos/terbitan/bab-37-matematik-asas/ km: https://astra-ai.co/km/videos/derivatives/chapter-311c891c/ lo: https://astra-ai.co/lo/videos/lo-d351a5ea/lo-311c891c/ es-AR: https://astra-ai.co/es-AR/videos/derivadas/capitulo-31/ zh-HK: https://astra-ai.co/zh-HK/videos/shuxue-kecheng-013/shuxue-zhang-037/ br: https://astra-ai.co/br/videos/derivadas/capitulo-matematica-037/ en-IN: https://astra-ai.co/en-IN/videos/derivatives/chapter-311c891c/ mx: https://astra-ai.co/mx/videos/derivadas/capitulo-31/ ms-BN: https://astra-ai.co/ms-BN/videos/terbitan/bab-311c891c/ es-PE: https://astra-ai.co/es-PE/videos/derivadas/capitulo-video-311c891c/ en-PH: https://astra-ai.co/en-PH/videos/derivatives/chapter-311c891c/ pt-TL: https://astra-ai.co/pt-TL/videos/derivadas/capitulo-de-video-031/ en: https://astra-ai.co/videos/derivatives/derivatives-of-other-elementary-functions/ --- # Derivatives of Other Elementary Functions > Derivatives of elementary functions cover exponentials, logarithms, and trigonometric functions. This chapter applies the standard differentiation rules. ## INTRODUCTION TO A WIDER RANGE OF FUNCTIONS In analysis, we deal with several types of basic functions that appear when finding derivatives. Besides polynomials, important functions include exponential, logarithmic, trigonometric functions, and their inverse forms. These are called elementary functions because they represent the building blocks of most more complex expressions. Each of them has its own rule for differentiation, based on the mathematical properties of the function. ## EXPONENTIAL FUNCTIONS For a function of the form f(x) = a^x, where 'a' is a positive number different from 1, the derivative is given as: f′(x) = a^x \* ln(a). A special case is the function f(x) = e^x, where e ≈ 2.718 (Euler's number), for which: f′(x) = e^x. This function remains unchanged upon differentiation. ## LOGARITHMIC FUNCTIONS The function f(x) = log\_a(x), where a > 0 and a ≠ 1, has the derivative: f′(x) = 1 / (x \* ln(a)). For the natural logarithm, i.e., f(x) = ln(x), a simplified rule applies: f′(x) = 1 / x, valid for x > 0. Logarithmic functions are not defined for negative values of x or for x = 0. ## TRIGONOMETRIC FUNCTIONS The derivatives of the basic trigonometric functions are: - d/dx (sin(x)) = cos(x) - d/dx (cos(x)) = -sin(x) - d/dx (tan(x)) = 1 / cos²(x) = sec²(x), for x ≠ (2k + 1)π/2 (where k is an integer) - d/dx (cot(x)) = -1 / sin²(x) = -csc²(x), for x ≠ kπ (where k is an integer) The tangent and cotangent functions have points where they are not defined, so their derivatives do not exist there. ## INVERSE TRIGONOMETRIC FUNCTIONS The inverse trigonometric functions have the following derivatives: - d/dx (arcsin(x)) = 1 / sqrt(1 − x²), for |x| < 1 - d/dx (arccos(x)) = −1 / sqrt(1 − x²), for |x| < 1 - d/dx (arctan(x)) = 1 / (1 + x²), for all real x These functions are used when solving problems involving compositions of functions or when there is a need to invert trigonometric expressions. ## CONCLUSION Different types of elementary functions require knowledge of their specific differentiation rules. Each type of function has its prescribed rule, based on the structure of the function and its domain. Understanding these rules is essential for more complex treatment of expressions in analysis.