derivative derivative function real variable measures sensitivity change quantity function value dependent variable determined quantity independent variable derivatives fundamental tool example derivative position moving object respect time object velocity measures quickly position object changes time advanced derivative function single variable chosen input value exists slope line function point line best linear function near input value reason derivative described rate change ratio change dependent variable independent variable derivatives functions real derivative linear transformation appropriate translation best linear original function matrix matrix represents linear transformation respect basis given choice independent dependent calculated terms partial derivatives respect independent real valued function matrix reduces gradient process finding derivative called reverse process called fundamental states integration integration constitute fundamental operations single variable derivative action computing derivative derivative function variable measure rate value function changes respect change variable called derivative respect real numbers plotted derivative slope point simplest case apart trivial case constant function linear function meaning divided line case real numbers slope given formula symbol delta change formula true formula formula follows formula gives exact value slope line function linear straight line change divided change varies method find exact value rate change given value idea illustrated figures rate change limit value ratio differences infinitely small distinct commonly derivative change denoted derivative respect written formula suggesting ratio quantities expression read derivative respect oral form lead confusion derivative respect function denoted read prime read prime case ambiguity variable implied incorrectly attributed newton rigorous definition common approach turn intuitive idea precise definition define derivative limit difference real numbers approach described let real valued function defined open neighborhood real number classical geometry line function unique line point meet meaning line pass straight derivative respect slope line slope line close slope line nearby point example lines called lines value close zero gives good slope line smaller values absolute value general better slope line difference values points divided difference values formula expression newton difference passing exact answer limit limit lines line limit difference approaches zero exists represent slope line limit defined derivative function formula limit exists said common derivative derivative property formula intuitive interpretation figure line gives best linear formula near small interpretation easiest settings difference causes division zero slope line found directly method instead define difference function formula slope line continuous function meaning curve gaps continuous function away limit exists meaning way choosing value makes continuous function function derivative equals practice existence continuous extension difference shown modifying cancel limit value small clear defined process long tedious complicated functions commonly process definition relative extension real numbers derivative real function real point defined shadow natural extension denoted derivative said exist shadow independent chosen example function derivative result established calculating limit approaches zero difference formula expression shows difference equals definition difference definition limit says difference need defined limit result letting zero meaning value tends small formula slope function point derivative generally similar shows derivative function continuity continuous example choose point let step function returns value returns different value greater equal derivative negative low step line steep tends zero slope tends infinity positive high step line slope zero consequently lines approach single slope limit difference exist function continuous point example absolute value function continuous positive slope line negative slope line negative seen function smooth point vertical instance function summary function derivative necessary function continuous continuity sufficient functions occur practice derivatives points point early history assumed continuous function points mild conditions example function function function true found example function continuous example known function proved set functions derivative point set space continuous functions informally means hardly continuous functions derivative point derivative function let function derivative point domain point derivative function sends point derivative function written called derivative function derivative derivative collects derivatives points domain derivative points domain function value equals defined called derivative function domain strictly smaller domain idea function functions derivative operator domain set functions derivatives point domain range set functions denote operator function function evaluated point definition derivative function comparison consider doubling function real valued function real number meaning takes numbers numbers formula operator defined individual numbers defined functions formula output function output evaluated point instance applied function doubling function named output function evaluated higher derivatives let function let derivative derivative written called second derivative similarly derivative second derivative exists written called derivative continuing process define exists derivative derivative derivative repeated derivatives called higher order derivatives derivative called derivative order represents position object time higher order derivatives physical interpretations second derivative derivative velocity definition object acceleration derivative defined fourth derivative defined function need derivative example continuous similarly derivative second derivative example let formula calculation shows function derivative formula twice absolute value function derivative zero similar examples function derivatives non negative order derivative function successive derivatives called times addition derivative continuous function said class stronger condition having derivatives example class function infinitely derivatives called infinitely smooth real line function infinitely standard rules degree differentiated times constant function subsequent derivatives zero particular exist smooth functions derivatives function point provide function near example twice formula sense formula infinitely beginning taylor series evaluated point point second derivative function changes sign called point point second derivative zero case point function fail exist case point function point function switches convex function concave function vice versa details derivatives introduced earliest commonly equation viewed functional relationship dependent independent derivative denoted formula thought higher derivatives expressed formula derivative respect multiple applications derivative operator example formula write derivative point different ways formula allows specify variable especially relevant partial makes chain rule easy remember formula referred prime common modern uses prime mark derivative function denoted simply similarly second derivatives denoted formula formula denote number derivatives point authors use roman place number formula formula yield derivative useful wish talk derivative function case newton newton called dot places dot function represent time derivative formula formula denote respectively second derivatives respect exclusively derivatives respect time arc length common physics differential differential geometry high order derivatives practice derivatives needed uses differential operator applied function derivative second derivative denoted derivative denoted dependent variable attached clarify independent variable written formula formula omitted variable understood instance variable present expression useful stating solving linear differential rules derivative function principle definition considering difference computing limit practice derivatives simple functions known derivatives functions easily rules obtaining derivatives complicated functions simpler ones rules basic functions derivative eventually require taking derivative common functions following incomplete list gives frequently functions single real variable derivatives derivatives powers formula real number formula function defined example formula formula derivative function defined positive rule implies zero constant rule stated functions formula formula formula formula functions formula formula formula functions formula formula formula rules combined functions cases complicated limit calculations direct application newton difference avoided rules basic rules following constant rule constant formula sum rule formula functions real numbers formula formula product rule formula functions special case rule includes fact formula formula constant formula constant rule rule formula functions chain rule formula formula example derivative formula formula second term chain rule product rule known derivatives elementary functions sin constant derivatives higher dimensions derivatives valued functions valued function real variable sends real numbers space valued function split coordinate functions meaning includes example curves coordinate functions real valued functions definition derivative applies derivative defined called coordinates derivatives coordinate functions formula formula limit exists derivative exists value valued function standard basis written assume derivative valued function retains property derivative formula basis constant useful example position particle time derivative velocity particle time partial derivatives suppose function depends variable instance formula family functions variable formula words value chooses function denoted function real number formula formula value chosen determines function sends formula expression constant variable function real variable consequently definition derivative function variable applies formula procedure performed choice assembling derivatives function gives function describes variation direction formula partial derivative respect rounded called partial derivative symbol distinguish letter pronounced der del partial instead general partial derivative function direction point defined formula difference held fixed choice fixed values determines function variable formula definition formula words different choices index family variable functions example expression shows partial derivatives reduces variable derivatives important example function case valued function domain space case partial derivative respect variable point partial derivatives define formula called gradient point domain gradient valued function takes point consequently gradient determines field directional derivatives real valued function partial derivatives measure variation direction coordinate axes example function partial derivatives measure variation direction direction directly measure variation direction diagonal line measured directional derivatives choose formula directional derivative direction point limit formula cases easier estimate directional derivative changing length turn problem directional derivative direction unit works suppose substitute difference difference formula times difference directional derivative respect furthermore taking limit tends zero taking limit tends zero property directional derivatives frequently considered unit partial derivatives exist continuous determine directional derivative direction formula formula consequence definition total derivative follows directional derivative linear meaning definition works function values definition applied component case directional derivative total derivative total differential matrix function open directional derivative chosen direction best linear point direction single directional derivative complete picture behavior total derivative gives complete picture considering directions starting linear formula holds formula like single variable derivative chosen error small possible derivative number expression product numbers higher dimensions impossible number number terms formula sense linear formula sense function sends denote function evaluated determine kind function notice linear formula rewritten formula notice choose approximate equation determines approximate equation determines approximate equation new formula assume small derivative varies continuously approximately equal right hand approximately zero left hand rewritten different way linear formula substituted linear formula implies formula suggests linear transformation space space fact possible precise measuring error assume error linear formula bounded constant times constant independent depends continuously adding appropriate error term approximate particular linear transformation small error term limit tend zero linear transformation define total derivative taking limit goes zero linear transformation variable fact derivative best linear expressed fact limit difference usual difference sense higher dimensions usually possible divide particular difference space lies lies domain furthermore derivative linear transformation different type object precise idea best linear necessary adapt different formula variable derivative problems disappear usual definition derivative manipulated derivative unique number formula equivalent formula limit function tends zero limit absolute value function tends zero formula adapted variable situation replacing absolute values norms definition total derivative unique linear transformation formula norm standard length norm standard length starting called written total derivative exists partial derivatives directional derivatives exist directional derivative direction write coordinate functions total derivative expressed partial derivatives matrix matrix called matrix formula existence total derivative strictly stronger existence partial derivatives partial derivatives exist continuous total derivative exists given depends continuously definition total derivative definition derivative variable real valued function real variable total derivative exists usual derivative exists matrix reduces matrix entry derivative matrix property approximately zero words formula changing statement function formula best linear total derivative function function way variable case total derivative function record information derivative single variable function instead total derivative gives function bundle source bundle target natural analog second higher order total derivatives linear transformation function bundle built repeatedly taking total derivative analog higher order derivative called jet linear transformation higher order derivatives reflect subtle geometric information described terms linear data function bundle bundle room base space directional derivatives jets capture higher order information arguments additional coordinates representing higher order changes direction space determined additional coordinates called jet bundle relation total derivative partial derivatives function paralleled relation order jet function partial derivatives order equal repeatedly taking total derivative higher versions derivative specialized order total derivative interpreted map formula takes point assigns element space linear maps best certain precise sense linear point diagonal map taylor series begun formula identified constant function components components linear concept derivative extended settings common thread derivative function point serves linear function point important derivative concerns complex functions complex functions domain complex numbers notion derivative function obtained replacing real complex definition identified writing complex number function certainly function sense partial derivatives exist true general complex derivative exists real derivative complex linear relations partial derivatives called functions concerns functions smooth speaking space near point space called space example smooth surface derivative differential map point linear map space space derivative function map definition fundamental differential geometry uses differential differential geometry defined maps infinite dimensional spaces spaces spaces directional derivative called derivative differential called derivative classical derivative functions way extending notion derivative continuous functions functions differentiated concept known weak derivative idea continuous functions larger space called space require function average properties derivative inspired introduction study similar objects example differential equivalent finite differences study differential unified finite differences time scale 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