Editing Calculus (section)

=== Differential calculus ===
{{Main|Differential calculus}}
[[File:Tangent line to a curve.svg|thumb|upright=1.35 |Tangent line at {{math|(''x''<sub>0</sub>, ''f''(''x''<sub>0</sub>))}}. The derivative {{math|''f′''(''x'')}} of a curve at a point is the slope (rise over run) of the line tangent to that curve at that point.]]

Differential calculus is the study of the definition, properties, and applications of the [[derivative]] of a function. The process of finding the derivative is called ''differentiation''. Given a function and a point in the domain, the derivative at that point is a way of encoding the small-scale behavior of the function near that point. By finding the derivative of a function at every point in its domain, it is possible to produce a new function, called the ''derivative function'' or just the ''derivative'' of the original function. In formal terms, the derivative is a [[linear operator]] which takes a function as its input and produces a second function as its output. This is more abstract than many of the processes studied in elementary algebra, where functions usually input a number and output another number. For example, if the doubling function is given the input three, then it outputs six, and if the squaring function is given the input three, then it outputs nine. The derivative, however, can take the squaring function as an input. This means that the derivative takes all the information of the squaring function—such as that two is sent to four, three is sent to nine, four is sent to sixteen, and so on—and uses this information to produce another function. The function produced by differentiating the squaring function turns out to be the doubling function.<ref name="TMU">{{Cite book |last1=Frautschi |first1=Steven C. |title=The Mechanical Universe: Mechanics and Heat |title-link=The Mechanical Universe |last2=Olenick |first2=Richard P. |last3=Apostol |first3=Tom M. |last4=Goodstein |first4=David L. |date=2007 |publisher=Cambridge University Press |isbn=978-0-521-71590-4 |edition=Advanced |location=Cambridge [Cambridgeshire] |oclc=227002144 |author-link=Steven Frautschi |author-link3=Tom M. Apostol |author-link4=David L. Goodstein}}</ref>{{Rp|32}}

In more explicit terms the "doubling function" may be denoted by {{math|''g''(''x'') {{=}} 2''x''}} and the "squaring function" by {{math|''f''(''x'') {{=}} ''x''<sup>2</sup>}}. The "derivative" now takes the function {{math|''f''(''x'')}}, defined by the expression "{{math|''x''<sup>2</sup>}}", as an input, that is all the information—such as that two is sent to four, three is sent to nine, four is sent to sixteen, and so on—and uses this information to output another function, the function {{math|''g''(''x'') {{=}} 2''x''}}, as will turn out.

In [[Lagrange's notation]], the symbol for a derivative is an [[apostrophe]]-like mark called a [[prime (symbol)|prime]]. Thus, the derivative of a function called {{math|''f''}} is denoted by {{math|''f′''}}, pronounced "f prime" or "f dash". For instance, if {{math|''f''(''x'') {{=}} ''x''<sup>2</sup>}} is the squaring function, then {{math|''f′''(''x'') {{=}} 2''x''}} is its derivative (the doubling function {{math|''g''}} from above).

If the input of the function represents time, then the derivative represents change concerning time. For example, if {{math|''f''}} is a function that takes time as input and gives the position of a ball at that time as output, then the derivative of {{math|''f''}} is how the position is changing in time, that is, it is the [[velocity]] of the ball.<ref name="TMU"/>{{Rp|18–20}}

If a function is [[linear function|linear]] (that is if the [[Graph of a function|graph]] of the function is a straight line), then the function can be written as {{math|''y'' {{=}} ''mx'' + ''b''}}, where {{math|''x''}} is the independent variable, {{math|''y''}} is the dependent variable, {{math|''b''}} is the ''y''-intercept, and:

:<math>m= \frac{\text{rise}}{\text{run}}= \frac{\text{change in } y}{\text{change in } x} = \frac{\Delta y}{\Delta x}.</math>

This gives an exact value for the slope of a straight line.<ref name=":4">{{Cite book |last1=Salas |first1=Saturnino L. |title=Calculus; one and several variables |last2=Hille |first2=Einar |date=1971 |publisher=Xerox College Pub. |location=Waltham, MA |oclc=135567}}</ref>{{Rp|page=6}} If the graph of the function is not a straight line, however, then the change in {{math|''y''}} divided by the change in {{math|''x''}} varies. Derivatives give an exact meaning to the notion of change in output concerning change in input. To be concrete, let {{math|''f''}} be a function, and fix a point {{math|''a''}} in the domain of {{math|''f''}}. {{math|(''a'', ''f''(''a''))}} is a point on the graph of the function. If {{math|''h''}} is a number close to zero, then {{math|''a'' + ''h''}} is a number close to {{math|''a''}}. Therefore, {{math|(''a'' + ''h'', ''f''(''a'' + ''h''))}} is close to {{math|(''a'', ''f''(''a''))}}. The slope between these two points is

:<math>m = \frac{f(a+h) - f(a)}{(a+h) - a} = \frac{f(a+h) - f(a)}{h}.</math>

This expression is called a ''[[difference quotient]]''. A line through two points on a curve is called a ''secant line'', so {{math|''m''}} is the slope of the secant line between {{math|(''a'', ''f''(''a''))}} and {{math|(''a'' + ''h'', ''f''(''a'' + ''h''))}}. The second line is only an approximation to the behavior of the function at the point {{math|'' a''}} because it does not account for what happens between {{math|'' a''}} and {{math|'' a'' + ''h''}}. It is not possible to discover the behavior at {{math|'' a''}} by setting {{math|''h''}} to zero because this would require [[dividing by zero]], which is undefined. The derivative is defined by taking the [[limit (mathematics)|limit]] as {{math|''h''}} tends to zero, meaning that it considers the behavior of {{math|''f''}} for all small values of {{math|''h''}} and extracts a consistent value for the case when {{math|''h''}} equals zero:

:<math>\lim_{h \to 0}{f(a+h) - f(a)\over{h}}.</math>

Geometrically, the derivative is the slope of the [[tangent line]] to the graph of {{math|''f''}} at {{math|''a''}}. The tangent line is a limit of secant lines just as the derivative is a limit of difference quotients. For this reason, the derivative is sometimes called the slope of the function {{math|''f''}}.<ref name=":4" />{{Rp|pages=61–63}}

Here is a particular example, the derivative of the squaring function at the input 3. Let {{math|''f''(''x'') {{=}} ''x''<sup>2</sup>}} be the squaring function.

[[File: Sec2tan.gif|thumb|upright=1.35|The derivative {{math|''f′''(''x'')}} of a curve at a point is the slope of the line tangent to that curve at that point. This slope is determined by considering the limiting value of the slopes of the second lines. Here the function involved (drawn in red) is {{math|''f''(''x'') {{=}} ''x''<sup>3</sup> − ''x''}}. The tangent line (in green) which passes through the point {{nowrap|(−3/2, −15/8)}} has a slope of 23/4. The vertical and horizontal scales in this image are different.]]

:<math>\begin{align}f'(3) &=\lim_{h \to 0}{(3+h)^2 - 3^2\over{h}} \\
&=\lim_{h \to 0}{9 + 6h + h^2 - 9\over{h}} \\
&=\lim_{h \to 0}{6h + h^2\over{h}} \\
&=\lim_{h \to 0} (6 + h) \\
&= 6
\end{align}
</math>

The slope of the tangent line to the squaring function at the point (3, 9) is 6, that is to say, it is going up six times as fast as it is going to the right. The limit process just described can be performed for any point in the domain of the squaring function. This defines the ''derivative function'' of the squaring function or just the ''derivative'' of the squaring function for short. A computation similar to the one above shows that the derivative of the squaring function is the doubling function.<ref name=":4" />{{Rp|page=63}}