In de wiskunde meet de afgeleide van een functie van een reële variabele de gevoeligheid voor verandering van de functiewaarde (outputwaarde) met betrekking tot een verandering in zijn argument (inputwaarde). Derivaten zijn een fundamenteel hulpmiddel bij calculus. De afgeleide van de positie van een bewegend object ten opzichte van de tijd is bijvoorbeeld de snelheid van het object: dit meet hoe snel de positie van het object verandert als de tijd verstrijkt.
De afgeleide van een functie van een enkele variabele bij een gekozen invoerwaarde, als deze bestaat, is de helling van de raaklijn naar de grafiek van de functie op dat punt. De raaklijn is de beste lineaire benadering van de functie in de buurt van die invoerwaarde. Om deze reden wordt de afgeleide vaak beschreven als de "instantane veranderingssnelheid", de verhouding tussen de instantane verandering in de afhankelijke variabele en die van de onafhankelijke variabele.
Derivaten kunnen worden gegeneraliseerd naar functies van verschillende reële variabelen. In deze generalisatie wordt de afgeleide geherinterpreteerd als een lineaire transformatie waarvan de grafiek (na een geschikte vertaling) de beste lineaire benadering is van de grafiek van de oorspronkelijke functie. De Jacobiaanse matrix is de matrix die deze lineaire transformatie vertegenwoordigt met betrekking tot de basis die wordt gegeven door de keuze van onafhankelijke en afhankelijke variabelen. Het kan worden berekend in termen van de partiële afgeleiden met betrekking tot de onafhankelijke variabelen. Voor een reële waarde van meerdere variabelen wordt de Jacobiaanse matrix gereduceerd tot de gradiëntvector.
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