Editing Binomial theorem (section)

=== Geometric explanation ===
[[File:binomial_theorem_visualisation.svg|thumb|300px|Visualisation of binomial expansion up to the 4th power]]
For positive values of {{mvar|a}} and {{mvar|b}}, the binomial theorem with {{math|1=''n'' = 2}} is the geometrically evident fact that a square of side {{math|''a'' + ''b''}} can be cut into a square of side {{mvar|a}}, a square of side {{mvar|b}}, and two rectangles with sides {{mvar|a}} and {{mvar|b}}. With {{math|1=''n'' = 3}}, the theorem states that a cube of side {{math|''a'' + ''b''}} can be cut into a cube of side {{mvar|a}}, a cube of side {{mvar|b}}, three {{math|''a'' × ''a'' × ''b''}} rectangular boxes, and three {{math|''a'' × ''b'' × ''b''}} rectangular boxes.

In [[calculus]], this picture also gives a geometric proof of the [[derivative]] <math>(x^n)'=nx^{n-1}:</math><ref name="barth2004">{{cite journal | last = Barth | first = Nils R.| title = Computing Cavalieri's Quadrature Formula by a Symmetry of the ''n''-Cube | doi = 10.2307/4145193 | jstor = 4145193 | journal = The American Mathematical Monthly | volume = 111| issue = 9| pages = 811–813 | date=2004}}</ref> if one sets <math>a=x</math> and <math>b=\Delta x,</math> interpreting {{mvar|b}} as an [[infinitesimal]] change in {{mvar|a}}, then this picture shows the infinitesimal change in the volume of an {{mvar|n}}-dimensional [[hypercube]], <math>(x+\Delta x)^n,</math> where the coefficient of the linear term (in <math>\Delta x</math>) is <math>nx^{n-1},</math> the area of the {{mvar|n}} faces, each of dimension {{math|''n'' &minus; 1}}:
<math display="block">(x+\Delta x)^n = x^n + nx^{n-1}\Delta x + \binom{n}{2}x^{n-2}(\Delta x)^2 + \cdots.</math>
Substituting this into the [[definition of the derivative]] via a [[difference quotient]] and taking limits means that the higher order terms, <math>(\Delta x)^2</math> and higher, become negligible, and yields the formula <math>(x^n)'=nx^{n-1},</math> interpreted as
:"the infinitesimal rate of change in volume of an {{mvar|n}}-cube as side length varies is the area of {{mvar|n}} of its {{math|(''n'' &minus; 1)}}-dimensional faces".
If one integrates this picture, which corresponds to applying the [[fundamental theorem of calculus]], one obtains [[Cavalieri's quadrature formula]], the integral <math>\textstyle{\int x^{n-1}\,dx = \tfrac{1}{n} x^n}</math> – see [[Cavalieri's quadrature formula#Proof|proof of Cavalieri's quadrature formula]] for details.<ref name="barth2004" />

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