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The differential was first introduced via an intuitive or heuristic definition by Isaac Newton and furthered by Gottfried Leibniz, who thought of the differential dy as an infinitely small (or infinitesimal) change in the value y of the function, corresponding to an infinitely small change dx in the function's argument x.
Similarly, the higher derivatives may be obtained inductively. While it is possible, with carefully chosen definitions, to interpret dy / dx as a quotient of differentials, this should not be done with the higher order forms. [17] However, an alternative Leibniz notation for differentiation for higher orders allows for this. [citation ...
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.
Later authors have assigned them other meanings, such as infinitesimals in non-standard analysis, or exterior derivatives. Commonly, dx is left undefined or equated with , while dy is assigned a meaning in terms of dx, via the equation =, which may also be written, e.g.
The derivatives in the table above are for when the range of the inverse secant is [,] and when the range of the inverse cosecant is [,]. It is common to additionally define an inverse tangent function with two arguments , arctan ( y , x ) . {\displaystyle \arctan(y,x).}
In Leibniz's notation, if x is a variable quantity, then dx denotes an infinitesimal change in the variable x. Thus, if y is a function of x, then the derivative of y with respect to x is often denoted dy/dx, which would otherwise be denoted (in the notation of Newton or Lagrange) ẏ or y ′.
Derivatives are frequently used to find the maxima and minima of a function. Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena . Derivatives and their generalizations appear in many fields of mathematics, such as complex analysis , functional analysis , differential geometry ...
The following derivatives are found by setting a variable y equal to the inverse trigonometric function that we wish to take the derivative of. Using implicit differentiation and then solving for dy/dx, the derivative of the inverse function is found in terms of y.