Tangent

tangent linetangentialtangentstangencytangentiallytangent pointtangent linestangent problemat a single pointintersection
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point.wikipedia
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Derivative

differentiationdifferentiablefirst derivative
More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point (c, f(c)) on the curve and has slope f(c), where f is the derivative of f.
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.

Differential calculus

differentialdifferentiationcalculus
These methods led to the development of differential calculus in the 17th century.
Geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point, provided that the derivative exists and is defined at that point.

Adequality

adequal
In the 1630s Fermat developed the technique of adequality to calculate tangents and other problems in analysis and used this to calculate tangents to the parabola.
Adequality is a technique developed by Pierre de Fermat in his treatise Methodus ad disquirendam maximam et minimam (a Latin treatise circulated in France c. 1636) to calculate maxima and minima of functions, tangents to curves, area, center of mass, least action, and other problems in calculus.

Method of normals

The technique of adeqality is similar to taking the difference between f(x+h) and f(x) and dividing by a power of h. Independently Descartes used his method of normals based on the observation that the radius of a circle is always normal to the circle itself.
In calculus, the method of normals was a technique invented by Descartes for finding normal and tangent lines to curves.

Apollonius of Perga

ApolloniusApollonius of PergeApollonian
In Apollonius work Conics (c.
Tangents are covered at the end of the book.

Parabola

parabolicparabolic curveparabolic arc
Circles, parabolas, hyperbolas and ellipses do not have any inflection point, but more complicated curves do have, like the graph of a cubic function, which has exactly one inflection point, or a sinusoid, which has two inflection points per each period of the sine.
Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane which is parallel to another plane that is tangential to the conical surface.

Euclid's Elements

ElementsEuclid's ''ElementsEuclid
Euclid makes several references to the tangent (ἐφαπτομένη ephaptoménē) to a circle in book III of the Elements (c.

Pierre de Fermat

FermatPierre FermatFermat, Pierre de
In the 1630s Fermat developed the technique of adequality to calculate tangents and other problems in analysis and used this to calculate tangents to the parabola.
In Methodus ad disquirendam maximam et minimam and in De tangentibus linearum curvarum, Fermat developed a method (adequality) for determining maxima, minima, and tangents to various curves that was equivalent to differential calculus.

Line (geometry)

linestraight linelines
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point.

Gottfried Wilhelm Leibniz

LeibnizGottfried LeibnizGottfried Wilhelm von Leibniz
Leibniz defined it as the line through a pair of infinitely close points on the curve. Further developments included those of John Wallis and Isaac Barrow, leading to the theory of Isaac Newton and Gottfried Leibniz.
Although the mathematical notion of function was implicit in trigonometric and logarithmic tables, which existed in his day, Leibniz was the first, in 1692 and 1694, to employ it explicitly, to denote any of several geometric concepts derived from a curve, such as abscissa, ordinate, tangent, chord, and the perpendicular.

Gilles de Roberval

Gilles Personne de RobervalRobervalde Roberval, Gilles
Roberval discovered a general method of drawing tangents, by considering a curve as described by a moving point whose motion is the resultant of several simpler motions.
Another of Roberval’s discoveries was a very general method of drawing tangents, by considering a curve as described by a moving point whose motion is the resultant of several simpler motions.

René-François de Sluse

François Walther de SluzeRené François Walter de SluseRenatus Franciscus Slusius
René-François de Sluse and Johannes Hudde found algebraic algorithms for finding tangents.
Sluse contributed to the development of calculus and this work focuses upon spirals, tangents, turning points and points of inflection.

Inflection point

inflection pointspoint of inflectioninflection
This old definition prevents inflection points from having any tangent.
If all extrema of f′ are isolated, then an inflection point is a point on the graph of f at which the tangent crosses the curve.

Secant line

secant
The intuitive notion that a tangent line "touches" a curve can be made more explicit by considering the sequence of straight lines (secant lines) passing through two points, A and B, those that lie on the function curve. The slope of the secant line passing through p and q is equal to the difference quotient
Secants may be used to approximate the tangent line to a curve, at some point

Slope

gradientslopesgradients
The slope of the secant line passing through p and q is equal to the difference quotient
As a generalization of this practical description, the mathematics of differential calculus defines the slope of a curve at a point as the slope of the tangent line at that point.

Exponential function

exponentialexponentiallyexp
Calculus provides rules for computing the derivatives of functions that are given by formulas, such as the power function, trigonometric functions, exponential function, logarithm, and their various combinations.
The slope of the tangent to the graph at each point is equal to its y-coordinate at that point, as implied by its derivative function (see above).

Isaac Barrow

BarrowBarrow, Isaac
Further developments included those of John Wallis and Isaac Barrow, leading to the theory of Isaac Newton and Gottfried Leibniz.
The geometrical lectures contain some new ways of determining the areas and tangents of curves.

Logarithm

logarithmsloglogarithmic function
Calculus provides rules for computing the derivatives of functions that are given by formulas, such as the power function, trigonometric functions, exponential function, logarithm, and their various combinations.
That is, the slope of the tangent touching the graph of the

Calculus

infinitesimal calculusdifferential and integral calculusclassical calculus
The question of finding the tangent line to a graph, or the tangent line problem, was one of the central questions leading to the development of calculus in the 17th century.
Geometrically, the derivative is the slope of the tangent line to the graph of

Normal (geometry)

normalnormal vectorsurface normal
For example, in two dimensions, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at the point.

Circle

circularcircles360 degrees
Circles, parabolas, hyperbolas and ellipses do not have any inflection point, but more complicated curves do have, like the graph of a cubic function, which has exactly one inflection point, or a sinusoid, which has two inflection points per each period of the sine.
The tangent line through a point P on the circle is perpendicular to the diameter passing through P.

Newton's method

Newton–Raphson methodNewton-RaphsonNewton–Raphson
-axis and the tangent of the graph of

Triangle

triangular{3}triangles
This is the case, for example, for a line passing through the vertex of a triangle and not intersecting it otherwise—where the tangent line does not exist for the reasons explained above.
The tangential triangle of a reference triangle (other than a right triangle) is the triangle whose sides are on the tangent lines to the reference triangle's circumcircle at its vertices.

René Descartes

DescartesCartesianRene Descartes
The technique of adeqality is similar to taking the difference between f(x+h) and f(x) and dividing by a power of h. Independently Descartes used his method of normals based on the observation that the radius of a circle is always normal to the circle itself.
Descartes' work provided the basis for the calculus developed by Newton and Leibniz, who applied infinitesimal calculus to the tangent line problem, thus permitting the evolution of that branch of modern mathematics.

Tangent lines to circles

common tangentBitangents to pairs of circlescommon internal tangent lines
Since the tangent line to a circle at a point P is perpendicular to the radius to that point, theorems involving tangent lines often involve radial lines and orthogonal circles.