Editing Tangent (section)

===Analytical approach===
[[Image:Graph of sliding derivative line.gif|right|thumb|upright=1.25|At each point, the moving line is always tangent to the [[curve]]. Its slope is the [[derivative]]; green marks positive derivative, red marks negative derivative and black marks zero derivative. The point (x,y) = (0,1) where the tangent intersects the curve, is not a [[Maxima and minima|max]], or a min, but is a [[point of inflection]]. (Note: the figure contains the incorrect labeling of 0,0 which should be 0,1)]]

The geometrical idea of the tangent line as the limit of secant lines serves as the motivation for analytical methods that are used to find tangent lines explicitly. 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. In the second book of his ''[[La Geometrie|Geometry]]'', [[René Descartes]]<ref>{{cite book |publisher=Open Court |page=95 |last=Descartes |first=René |title=The Geometry of René Descartes |year=1954 |orig-year=1637 |url=https://archive.org/details/geometryofrene00desc/page/95/ |translator1-last= Smith |translator1-first=David Eugene |translator1-link=David Eugene Smith |translator2-last=Latham |translator2-first=Marcia L. }}</ref> [[s:fr:Page:Descartes La Géométrie.djvu/52|said]] of the problem of constructing the tangent to a curve, "And I dare say that this is not only the most useful and most general problem in geometry that I know, but even that I have ever desired to know".<ref>{{cite journal |author=R. E. Langer |date=October 1937 |title=Rene Descartes |journal=[[American Mathematical Monthly]] |volume=44 |issue=8 |pages=495–512 |publisher=Mathematical Association of America |doi=10.2307/2301226 |jstor=2301226}}</ref>

====Intuitive description====
Suppose that a curve is given as the graph of a [[function (mathematics)|function]], ''y'' = ''f''(''x''). To find the tangent line at the point ''p'' = (''a'', ''f''(''a'')), consider another nearby point ''q'' = (''a'' + ''h'', ''f''(''a'' + ''h'')) on the curve. The [[slope]] of the [[secant line]] passing through ''p'' and ''q'' is equal to the [[difference quotient]]

<math display="block">\frac{f(a+h)-f(a)}{h}.</math>

As the point ''q'' approaches ''p'', which corresponds to making ''h'' smaller and smaller, the difference quotient should approach a certain limiting value ''k'', which is the slope of the tangent line at the point ''p''. If ''k'' is known, the equation of the tangent line can be found in the point-slope form:

<math display="block"> y-f(a) = k(x-a).\,</math>

====More rigorous description====
To make the preceding reasoning rigorous, one has to explain what is meant by the difference quotient approaching a certain limiting value ''k''. The precise mathematical formulation was given by [[Augustin-Louis Cauchy|Cauchy]] in the 19th century and is based on the notion of [[limit of a function|limit]]. Suppose that the graph does not have a break or a sharp edge at ''p'' and it is neither plumb nor too wiggly near ''p''. Then there is a unique value of ''k'' such that, as ''h'' approaches 0, the difference quotient gets closer and closer to ''k'', and the distance between them becomes negligible compared with the size of ''h'', if ''h'' is small enough. This leads to the definition of the slope of the tangent line to the graph as the limit of the difference quotients for the function ''f''. This limit is the [[Derivative#Definition via difference quotients|derivative]] of the function ''f'' at ''x'' = ''a'', denoted ''f''&nbsp;′(''a''). Using derivatives, the equation of the tangent line can be stated as follows:

: <math> y=f(a)+f'(a)(x-a).\,</math>

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. Thus, equations of the tangents to graphs of all these functions, as well as many others, can be found by the methods of calculus.

====How the method can fail====
Calculus also demonstrates that there are functions and points on their graphs for which the limit determining the slope of the tangent line does not exist. For these points the function ''f'' is ''non-differentiable''. There are two possible reasons for the method of finding the tangents based on the limits and derivatives to fail: either the geometric tangent exists, but it is a vertical line, which cannot be given in the point-slope form since it does not have a slope, or the graph exhibits one of three behaviors that precludes a geometric tangent.

The graph ''y'' = ''x''<sup>1/3</sup> illustrates the first possibility:  here the difference quotient at ''a'' = 0 is equal to ''h''<sup>1/3</sup>/''h'' = ''h''<sup>−2/3</sup>, which becomes very large as ''h'' approaches 0. This curve has a tangent line at the origin that is vertical.

The graph ''y'' = ''x''<sup>2/3</sup> illustrates another possibility: this graph has a ''[[Cusp (singularity)|cusp]]'' at the origin. This means that, when ''h'' approaches 0, the difference quotient at ''a'' = 0 approaches plus or minus infinity depending on the sign of ''x''. Thus both branches of the curve are near to the half vertical line for which ''y''=0, but none is near to the negative part of this line. Basically, there is no tangent at the origin in this case, but in some context one may consider this line as a tangent, and even, in [[algebraic geometry]], as a ''double tangent''.

The graph ''y'' = |''x''| of the [[absolute value]] function consists of two straight lines with different slopes joined at the origin. As a point ''q'' approaches the origin from the right, the secant line always has slope 1. As a point ''q'' approaches the origin from the left, the secant line always has slope −1. Therefore, there is no unique tangent to the graph at the origin. Having two different (but finite) slopes is called a ''corner''.

Finally, since differentiability implies continuity, the [[Contraposition|contrapositive]] states ''discontinuity'' implies non-differentiability. Any such jump or point discontinuity will have no tangent line. This includes cases where one slope approaches positive infinity while the other approaches negative infinity, leading to an infinite jump discontinuity