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# Third derivative

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 Title: Third derivative Author: World Heritage Encyclopedia Language: English Subject: Collection: Differential Calculus Publisher: World Heritage Encyclopedia Publication Date:

### Third derivative

In calculus, a branch of mathematics, the third derivative is the rate at which the second derivative, or the rate of change of the rate of change, is changing, used to define aberrancy.[1] The third derivative of a function y=f(x) can be denoted by

Other notations can be used, but the above are the most common.

## Contents

• Mathematical definitions 1
• Applications in geometry 2
• Applications in physics 3
• Economic example 4
• References 6

## Mathematical definitions

Let f(x)=x^4. Then f'(x)=4x^3, and f''(x)=12x^2. Therefore, the third derivative of f(x) is, in this case,

f'''(x)=24x

or, using Leibniz notation,

\frac{d^3}{dx^3}[x^4]=24x

Now for a more general definition. Let f(x) be any function of x. Then the third derivative of f(x) is given by the following:

\frac{d^3}{dx^3}[f(x)]=\frac{d}{dx}[f''(x)]

The third derivative is the rate at which the second derivative (f''(x)) is changing.

## Applications in geometry

In differential geometry, the torsion of a curve — a fundamental property of curves in three dimensions — is computed using third derivatives of coordinate functions (or the position vector) describing the curve.[2]

## Applications in physics

In physics, particularly kinematics, jerk is defined as the third derivative of the position function of an object. It is basically the rate at which acceleration is changing. In mathematical terms:

\bold{j}(t)=\frac{d^3\bold{r}}{dt^3}

where j(t) is the jerk function with respect to time, and r(t) is the position function of the object with respect to time.

## Economic example

U.S. President Richard Nixon, when campaigning for a second term in office announced that the rate of increase of inflation was decreasing, which has been noted as "the first time a sitting president used the third derivative to advance his case for reelection."[3] Since inflation is itself a derivative—the rate at which the purchasing power of money decreases—then the rate of increase of inflation is the derivative of inflation, or the second derivative of the function of purchasing power of money with respect to time. Stating that a function is decreasing is equivalent to stating that its derivative is negative, so Nixon's statement is that the second derivative of inflation—or the third derivative of purchasing power—is negative.

Nixon's statement allowed for the rate of inflation to increase, however, so his statement was not as indicative of stable prices as it sounds.require('Module:No globals')

local p = {}

-- articles in which traditional Chinese preceeds simplified Chinese local t1st = { ["228 Incident"] = true, ["Chinese calendar"] = true, ["Lippo Centre, Hong Kong"] = true, ["Republic of China"] = true, ["Republic of China at the 1924 Summer Olympics"] = true, ["Taiwan"] = true, ["Taiwan (island)"] = true, ["Taiwan Province"] = true, ["Wei Boyang"] = true, }

-- the labels for each part local labels = { ["c"] = "Chinese", ["s"] = "simplified Chinese", ["t"] = "traditional Chinese", ["p"] = "pinyin", ["tp"] = "Tongyong Pinyin", ["w"] = "Wade–Giles", ["j"] = "Jyutping", ["cy"] = "Cantonese Yale", ["poj"] = "Pe̍h-ōe-jī", ["zhu"] = "Zhuyin Fuhao", ["l"] = "literally", }

-- article titles for wikilinks for each part local wlinks = { ["c"] = "Chinese language", ["s"] = "simplified Chinese characters", ["t"] = "traditional Chinese characters", ["p"] = "pinyin", ["tp"] = "Tongyong Pinyin", ["w"] = "Wade–Giles", ["j"] = "Jyutping", ["cy"] = "Yale romanization of Cantonese", ["poj"] = "Pe̍h-ōe-jī", ["zhu"] = "Bopomofo", }

-- for those parts which are to be treated as languages their ISO code local ISOlang = { ["c"] = "zh", ["t"] = "zh-Hant", ["s"] = "zh-Hans", ["p"] = "zh-Latn-pinyin", ["tp"] = "zh-Latn", ["w"] = "zh-Latn-wadegile", ["j"] = "yue-jyutping", ["cy"] = "yue", ["poj"] = "hak", ["zhu"] = "zh-Bopo", }

local italic = { ["p"] = true, ["tp"] = true, ["w"] = true, ["j"] = true, ["cy"] = true, ["poj"] = true, } -- Categories for different kinds of Chinese text local cats = { ["c"] = "", ["s"] = "", ["t"] = "", }

function p.Zh(frame) -- load arguments module to simplify handling of args local getArgs = require('Module:Arguments').getArgs local args = getArgs(frame) return p._Zh(args) end function p._Zh(args) local uselinks = not (args["links"] == "no") -- whether to add links local uselabels = not (args["labels"] == "no") -- whether to have labels local capfirst = args["scase"] ~= nil

        local t1 = false -- whether traditional Chinese characters go first
local j1 = false -- whether Cantonese Romanisations go first
local testChar
if (args["first"]) then
for testChar in mw.ustring.gmatch(args["first"], "%a+") do
if (testChar == "t") then
t1 = true
end
if (testChar == "j") then
j1 = true
end
end
end
if (t1 == false) then
local title = mw.title.getCurrentTitle()
t1 = t1st[title.text] == true
end


-- based on setting/preference specify order local orderlist = {"c", "s", "t", "p", "tp", "w", "j", "cy", "poj", "zhu", "l"} if (t1) then orderlist[2] = "t" orderlist[3] = "s" end if (j1) then orderlist[4] = "j" orderlist[5] = "cy" orderlist[6] = "p" orderlist[7] = "tp" orderlist[8] = "w" end -- rename rules. Rules to change parameters and labels based on other parameters if args["hp"] then -- hp an alias for p ([hanyu] pinyin) args["p"] = args["hp"] end if args["tp"] then -- if also Tongyu pinyin use full name for Hanyu pinyin labels["p"] = "Hanyu Pinyin" end if (args["s"] and args["s"] == args["t"]) then -- Treat simplified + traditional as Chinese if they're the same args["c"] = args["s"] args["s"] = nil args["t"] = nil elseif (not (args["s"] and args["t"])) then -- use short label if only one of simplified and traditional labels["s"] = labels["c"] labels["t"] = labels["c"] end local body = "" -- the output string local params -- for creating HTML spans local label -- the label, i.e. the bit preceeding the supplied text local val -- the supplied text -- go through all possible fields in loop, adding them to the output for i, part in ipairs(orderlist) do if (args[part]) then -- build label label = "" if (uselabels) then label = labels[part] if (capfirst) then label = mw.language.getContentLanguage():ucfirst(

## References

-- Module:Hatnote -- -- -- -- This module produces hatnote links and links to related articles. It -- -- implements the and meta-templates and includes -- -- helper functions for other Lua hatnote modules. --

local libraryUtil = require('libraryUtil') local checkType = libraryUtil.checkType local mArguments -- lazily initialise Module:Arguments local yesno -- lazily initialise Module:Yesno

local p = {}

-- Helper functions

local function getArgs(frame) -- Fetches the arguments from the parent frame. Whitespace is trimmed and -- blanks are removed. mArguments = require('Module:Arguments') return mArguments.getArgs(frame, {parentOnly = true}) end

local function removeInitialColon(s) -- Removes the initial colon from a string, if present. return s:match('^:?(.*)') end

function p.findNamespaceId(link, removeColon) -- Finds the namespace id (namespace number) of a link or a pagename. This -- function will not work if the link is enclosed in double brackets. Colons -- are trimmed from the start of the link by default. To skip colon -- trimming, set the removeColon parameter to true. checkType('findNamespaceId', 1, link, 'string') checkType('findNamespaceId', 2, removeColon, 'boolean', true) if removeColon ~= false then link = removeInitialColon(link) end local namespace = link:match('^(.-):') if namespace then local nsTable = mw.site.namespaces[namespace] if nsTable then return nsTable.id end end return 0 end

function p.formatPages(...) -- Formats a list of pages using formatLink and returns it as an array. Nil -- values are not allowed. local pages = {...} local ret = {} for i, page in ipairs(pages) do ret[i] = p._formatLink(page) end return ret end

function p.formatPageTables(...) -- Takes a list of page/display tables and returns it as a list of -- formatted links. Nil values are not allowed. local pages = {...} local links = {} for i, t in ipairs(pages) do checkType('formatPageTables', i, t, 'table') local link = t[1] local display = t[2] links[i] = p._formatLink(link, display) end return links end

function p.makeWikitextError(msg, helpLink, addTrackingCategory) -- Formats an error message to be returned to wikitext. If -- addTrackingCategory is not false after being returned from -- Module:Yesno, and if we are not on a talk page, a tracking category -- is added. checkType('makeWikitextError', 1, msg, 'string') checkType('makeWikitextError', 2, helpLink, 'string', true) yesno = require('Module:Yesno') local title = mw.title.getCurrentTitle() -- Make the help link text. local helpText if helpLink then helpText = ' (help)' else helpText = end -- Make the category text. local category if not title.isTalkPage and yesno(addTrackingCategory) ~= false then category = 'Hatnote templates with errors' category = string.format( '%s:%s', mw.site.namespaces[14].name, category ) else category = end return string.format( '%s', msg, helpText, category ) end

-- Format link -- -- Makes a wikilink from the given link and display values. Links are escaped -- with colons if necessary, and links to sections are detected and displayed -- with " § " as a separator rather than the standard MediaWiki "#". Used in -- the template.

function p.formatLink(frame) local args = getArgs(frame) local link = args[1] local display = args[2] if not link then return p.makeWikitextError( 'no link specified', 'Template:Format hatnote link#Errors', args.category ) end return p._formatLink(link, display) end

function p._formatLink(link, display) -- Find whether we need to use the colon trick or not. We need to use the -- colon trick for categories and files, as otherwise category links -- categorise the page and file links display the file. checkType('_formatLink', 1, link, 'string') checkType('_formatLink', 2, display, 'string', true) link = removeInitialColon(link) local namespace = p.findNamespaceId(link, false) local colon if namespace == 6 or namespace == 14 then colon = ':' else colon = end -- Find whether a faux display value has been added with the | magic -- word. if not display then local prePipe, postPipe = link:match('^(.-)|(.*)$') link = prePipe or link display = postPipe end -- Find the display value. if not display then local page, section = link:match('^(.-)#(.*)$') if page then display = page .. ' § ' .. section end end -- Assemble the link. if display then return string.format('%s', colon, link, display) else return string.format('%s%s', colon, link) end end

-- Hatnote -- -- Produces standard hatnote text. Implements the template.

function p.hatnote(frame) local args = getArgs(frame) local s = args[1] local options = {} if not s then return p.makeWikitextError( 'no text specified', 'Template:Hatnote#Errors', args.category ) end options.extraclasses = args.extraclasses options.selfref = args.selfref return p._hatnote(s, options) end

function p._hatnote(s, options) checkType('_hatnote', 1, s, 'string') checkType('_hatnote', 2, options, 'table', true) local classes = {'hatnote'} local extraclasses = options.extraclasses local selfref = options.selfref if type(extraclasses) == 'string' then classes[#classes + 1] = extraclasses end if selfref then classes[#classes + 1] = 'selfref' end return string.format( '
%s
', table.concat(classes, ' '), s )

end

return p-------------------------------------------------------------------------------- -- Module:Hatnote -- -- -- -- This module produces hatnote links and links to related articles. It -- -- implements the and meta-templates and includes -- -- helper functions for other Lua hatnote modules. --

local libraryUtil = require('libraryUtil') local checkType = libraryUtil.checkType local mArguments -- lazily initialise Module:Arguments local yesno -- lazily initialise Module:Yesno

local p = {}

-- Helper functions

local function getArgs(frame) -- Fetches the arguments from the parent frame. Whitespace is trimmed and -- blanks are removed. mArguments = require('Module:Arguments') return mArguments.getArgs(frame, {parentOnly = true}) end

local function removeInitialColon(s) -- Removes the initial colon from a string, if present. return s:match('^:?(.*)') end

function p.findNamespaceId(link, removeColon) -- Finds the namespace id (namespace number) of a link or a pagename. This -- function will not work if the link is enclosed in double brackets. Colons -- are trimmed from the start of the link by default. To skip colon -- trimming, set the removeColon parameter to true. checkType('findNamespaceId', 1, link, 'string') checkType('findNamespaceId', 2, removeColon, 'boolean', true) if removeColon ~= false then link = removeInitialColon(link) end local namespace = link:match('^(.-):') if namespace then local nsTable = mw.site.namespaces[namespace] if nsTable then return nsTable.id end end return 0 end

function p.formatPages(...) -- Formats a list of pages using formatLink and returns it as an array. Nil -- values are not allowed. local pages = {...} local ret = {} for i, page in ipairs(pages) do ret[i] = p._formatLink(page) end return ret end

function p.formatPageTables(...) -- Takes a list of page/display tables and returns it as a list of -- formatted links. Nil values are not allowed. local pages = {...} local links = {} for i, t in ipairs(pages) do checkType('formatPageTables', i, t, 'table') local link = t[1] local display = t[2] links[i] = p._formatLink(link, display) end return links end

function p.makeWikitextError(msg, helpLink, addTrackingCategory) -- Formats an error message to be returned to wikitext. If -- addTrackingCategory is not false after being returned from -- Module:Yesno, and if we are not on a talk page, a tracking category -- is added. checkType('makeWikitextError', 1, msg, 'string') checkType('makeWikitextError', 2, helpLink, 'string', true) yesno = require('Module:Yesno') local title = mw.title.getCurrentTitle() -- Make the help link text. local helpText if helpLink then helpText = ' (help)' else helpText = end -- Make the category text. local category if not title.isTalkPage and yesno(addTrackingCategory) ~= false then category = 'Hatnote templates with errors' category = string.format( '%s:%s', mw.site.namespaces[14].name, category ) else category = end return string.format( '%s', msg, helpText, category ) end

-- Format link -- -- Makes a wikilink from the given link and display values. Links are escaped -- with colons if necessary, and links to sections are detected and displayed -- with " § " as a separator rather than the standard MediaWiki "#". Used in -- the template.

function p.formatLink(frame) local args = getArgs(frame) local link = args[1] local display = args[2] if not link then return p.makeWikitextError( 'no link specified', 'Template:Format hatnote link#Errors', args.category ) end return p._formatLink(link, display) end

function p._formatLink(link, display) -- Find whether we need to use the colon trick or not. We need to use the -- colon trick for categories and files, as otherwise category links -- categorise the page and file links display the file. checkType('_formatLink', 1, link, 'string') checkType('_formatLink', 2, display, 'string', true) link = removeInitialColon(link) local namespace = p.findNamespaceId(link, false) local colon if namespace == 6 or namespace == 14 then colon = ':' else colon = end -- Find whether a faux display value has been added with the | magic -- word. if not display then local prePipe, postPipe = link:match('^(.-)|(.*)$') link = prePipe or link display = postPipe end -- Find the display value. if not display then local page, section = link:match('^(.-)#(.*)$') if page then display = page .. ' § ' .. section end end -- Assemble the link. if display then return string.format('%s', colon, link, display) else return string.format('%s%s', colon, link) end end

-- Hatnote -- -- Produces standard hatnote text. Implements the template.

function p.hatnote(frame) local args = getArgs(frame) local s = args[1] local options = {} if not s then return p.makeWikitextError( 'no text specified', 'Template:Hatnote#Errors', args.category ) end options.extraclasses = args.extraclasses options.selfref = args.selfref return p._hatnote(s, options) end

function p._hatnote(s, options) checkType('_hatnote', 1, s, 'string') checkType('_hatnote', 2, options, 'table', true) local classes = {'hatnote'} local extraclasses = options.extraclasses local selfref = options.selfref if type(extraclasses) == 'string' then classes[#classes + 1] = extraclasses end if selfref then classes[#classes + 1] = 'selfref' end return string.format( '
%s
', table.concat(classes, ' '), s )

end

return p
1. ^
2. ^
3. ^
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