Editing Differential calculus (section)

== Derivative ==
{{main|Derivative}}
[[File:Curve with tangent line.png|alt=|thumb|The graph of an arbitrary function <math>y=f(x)</math>. The orange line is tangent to <math>x=a</math>, meaning at that exact point, the slope of the curve and the straight line are the same.]]
[[File:Tangent function animation.gif|thumb|300px|The derivative at different points of a differentiable function]]
The derivative of <math>f(x)</math> at the point <math>x=a</math> is the [[slope]] of the tangent to <math>(a,f(a))</math>.<ref>{{Cite book|last=Alcock|first=Lara|title=How to Think about Analysis|publisher=Oxford University Press|year=2016|isbn=978-0-19-872353-0|location=New York|pages=155–157}}</ref> In order to gain an intuition for this, one must first be familiar with finding the slope of a linear equation, written in the form <math>y=mx+b</math>. The slope of an equation is its steepness. It can be found by picking any two points and dividing the change in <math>y</math> by the change in <math>x</math>, meaning that <math>\text{slope } =\frac{\text{ change in }y}{\text{change in }x}</math>. For, the graph of <math>y=-2x+13</math> has a slope of <math>-2</math>, as shown in the diagram below:

[[File:The graph of y=-2x+13.png|alt=|none|thumb|The graph of <math>y=-2x+13</math>]]
:<math>\frac{\text{change in }y}{\text{change in }x}=\frac{-6}{+3}=-2</math>

For brevity, <math>\frac{\text{change in }y}{\text{change in }x}</math> is often written as <math>\frac{\Delta y}{\Delta x}</math>, with <math>\Delta</math> being the Greek letter delta, meaning 'change in'. The slope of a linear equation is constant, meaning that the steepness is the same everywhere. However, many graphs such as <math>y=x^2</math> vary in their steepness. This means that you can no longer pick any two arbitrary points and compute the slope. Instead, the slope of the graph can be computed by considering the tangent line—a line that 'just touches' a particular point.{{efn|This is not a formal definition of what a tangent line is. The definition of the derivative as a limit makes rigorous this notion of tangent line.}} The slope of a curve at a particular point is equal to the slope of the tangent to that point. For example, <math>y=x^2</math> has a slope of <math>4</math> at <math>x=2</math> because the slope of the tangent line to that point is equal to <math>4</math>:

[[File:Graph of the function f(x)=x^2 with a tangent line drawn to (2,4).png|thumb|none|The graph of <math>y=x^2</math>, with a straight line that is tangent to <math>(2,4)</math>. The slope of the tangent line is equal to <math>4</math>. (The axes of the graph do not use a 1:1 scale.)]]


The derivative of a [[Function (mathematics)|function]] is then simply the slope of this tangent line.{{efn|Though the technical definition of a [[Function (mathematics)|function]] is somewhat involved, it is easy to appreciate what a function is intuitively. A function takes an input and produces an output. For example, the function <math>f(x)=x^2</math> takes a number and squares it. The number that the function performs an operation on is often represented using the letter <math>x</math>, but there is no difference whatsoever between writing <math>f(x)=x^2</math> and writing <math>f(y)=y^2</math>. For this reason, <math>x</math> is often described as a 'dummy variable'.}} Even though the tangent line only touches a single point at the point of tangency, it can be approximated by a line that goes through two points. This is known as a [[secant line]]. If the two points that the secant line goes through are close together, then the secant line closely resembles the tangent line, and, as a result, its slope is also very similar:

[[File:Tangent line versus secant line.png|thumb|none|The dotted line goes through the points <math>(2,4)</math> and <math>(3,9)</math>, which both lie on the curve <math>y=x^2</math>. Because these two points are fairly close together, the dotted line and tangent line have a similar slope. As the two points become closer together, the error produced by the secant line becomes vanishingly small.]]


The advantage of using a secant line is that its slope can be calculated directly. Consider the two points on the graph <math>(x,f(x))</math> and <math>(x+\Delta x,f(x+\Delta x))</math>, where <math>\Delta x</math> is a small number. As before, the slope of the line passing through these two points can be calculated with the formula <math>\text{slope } = \frac{\Delta y}{\Delta x}</math>. This gives

:<math>\text{slope} = \frac{f(x+\Delta x)-f(x)}{\Delta x}</math>

As <math>\Delta x</math> gets closer and closer to <math>0</math>, the slope of the secant line gets closer and closer to the slope of the tangent line. This is formally written as

:<math>\lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}</math>

The above expression means 'as <math>\Delta x</math> gets closer and closer to 0, the slope of the secant line gets closer and closer to a certain value'. The value that is being approached is the derivative of <math>f(x)</math>; this can be written as <math>f'(x)</math>. If <math>y=f(x)</math>, the derivative can also be written as <math>\frac{dy}{dx}</math>, with <math>d</math> representing an [[infinitesimal]] change. For example, <math>dx</math> represents an infinitesimal change in x.{{efn|The term infinitesimal can sometimes lead people to wrongly believe there is an 'infinitely small number'—i.e. a positive real number that is smaller than any other real number. In fact, the term 'infinitesimal' is merely a shorthand for a limiting process. For this reason, <math>\frac{dy}{dx}</math> is not a fraction—rather, it is the limit of a fraction.}} In summary, if <math>y=f(x)</math>, then the derivative of <math>f(x)</math> is

:<math>
\frac{dy}{dx}=f'(x)=\lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}
</math>

provided such a limit exists.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Derivative|url=https://mathworld.wolfram.com/Derivative.html|access-date=2020-07-26|website=mathworld.wolfram.com|language=en}}</ref>{{efn|Not every function can be differentiated, hence why the definition only applies if 'the limit exists'. For more information, see the  Wikipedia article on [[differentiability]].}}  We have thus succeeded in properly defining the derivative of a function, meaning that the 'slope of the tangent line' now has a precise mathematical meaning. Differentiating a function using the above definition is known as differentiation from first principles. Here is a proof, using differentiation from first principles, that the derivative of <math>y=x^2</math> is <math>2x</math>:

:<math>
\begin{align}
\frac{dy}{dx}&=\lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x} \\
&= \lim_{\Delta x \to 0}\frac{(x+\Delta x)^2-x^2}{\Delta x} \\
&= \lim_{\Delta x \to 0}\frac{x^2+2x\Delta x+(\Delta x)^2-x^2}{\Delta x} \\
&= \lim_{\Delta x \to 0}\frac{2x\Delta x+(\Delta x)^2}{\Delta x} \\
&= \lim_{\Delta x \to 0}2x+\Delta x \\
\end{align}
</math>

As <math>\Delta x</math> approaches <math>0</math>, <math>2x+\Delta x</math> approaches <math>2x</math>. Therefore, <math>\frac{dy}{dx}=2x</math>. This proof can be generalised to show that <math>\frac{d(ax^n)}{dx}=anx^{n-1}</math> if <math>a</math> and <math>n</math> are [[Constant (mathematics)|constants]]. This is known as the [[power rule]]. For example, <math>\frac{d}{dx}(5x^4)=5(4)x^3=20x^3</math>. However, many other functions cannot be differentiated as easily as [[polynomial|polynomial functions]], meaning that sometimes further techniques are needed to find the derivative of a function. These techniques include the [[chain rule]], [[product rule]], and [[quotient rule]]. Other functions cannot be differentiated at all, giving rise to the concept of [[differentiability]].

A closely related concept to the derivative of a function is its [[differential (mathematics)|differential]]. When {{math|''x''}} and {{math|''y''}} are real variables, the derivative of {{math|''f''}} at {{math|''x''}} is the slope of the tangent line to the graph of {{math|''f''}} at {{math|''x''}}. Because the source and target of {{math|''f''}} are one-dimensional, the derivative of {{math|''f''}} is a real number. If {{math|''x''}} and {{math|''y''}} are vectors, then the best linear approximation to the graph of {{math|''f''}} depends on how {{math|''f''}} changes in several directions at once. Taking the best linear approximation in a single direction determines a [[partial derivative]], which is usually denoted {{math|{{sfrac|∂''y''|∂''x''}}}}. The linearization of {{math|''f''}} in all directions at once is called the [[total derivative]].