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{{Calculus |Differential}}
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This is a summary of '''differentiation rules''', that is, rules for computing the [[derivative]] of a [[function (mathematics)|function]] in [[calculus]].
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== Elementary rules of differentiation ==
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Unless otherwise stated, all functions are functions of [[real number|real numbers ('''R''')]] that return real values; although more generally, the formulae below apply wherever they are [[well defined]]<ref>''Calculus (5th edition)'', F. Ayres, E. Mendelson, Schuam's Outline Series, 2009, ISBN 978-0-07-150861-2.</ref><ref>''Advanced Calculus (3rd edition)'', R. Wrede, M.R. Spiegel, Schuam's Outline Series, 2010, ISBN 978-0-07-162366-7.</ref>—including [[complex number|complex numbers ('''C''')]].<ref>''Complex Variables'', M.R. Speigel, S. Lipschutz, J.J. Schiller, D. Spellman, Schaum's Outlines Series, McGraw Hill (USA), 2009, ISBN 978-0-07-161569-3</ref>
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===Differentiation is linear===
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{{main|Linearity of differentiation}}


For any functions ''f'' and ''g'' and any real numbers ''a'' and ''b'' the derivative of the function {{nowrap|1=''h''(''x'') = ''af''(''x'') + ''bg''(''x'')}} with respect to ''x'' is
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:<math> h'(x) = a f'(x) + b g'(x).\, </math>
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In [[Leibniz's notation]] this is written as:
:<math> \frac{d(af+bg)}{dx} = a\frac{df}{dx} +b\frac{dg}{dx}.</math>


Special cases include:
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* ''The [[Constant factor rule in differentiation|constant division
 
rule]]''
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:<math>(af)' = af' \,</math>
 
* ''The [[Sum rule in differentiation|sum rule]]''
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:<math>(f + g)' = f' + g'\,</math>
 
* ''The subtraction rule''
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===The product rule===
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{{main|Product rule}}
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For the functions ''f'' and ''g'', the derivative of the function ''h''(''x'') = ''f''(''x'') ''g''(''x'')
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with respect to ''x'' is
:<math> h'(x) = f'(x) g(x) + f(x) g'(x).\, </math>
In Leibniz's notation this is written
:<math>\frac{d(fg)}{dx} = \frac{df}{dx} g + f \frac{dg}{dx}.</math>


===The chain rule===
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{{main|Chain rule}}
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The derivative of the function of a function ''h''(''x'') = ''f''(''g''(''x'')) with respect to ''x'' is
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:<math> h'(x) = f'(g(x)) g'(x).\, </math>
 
In Leibniz's notation this is written as:
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:<math>\frac{dh}{dx} = \frac{df(g(x))}{dg(x)} \frac{dg(x)}{dx}.\,</math>
 
However, by relaxing the interpretation of ''h'' as a function, this is often simply written
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:<math>\frac{dh}{dx} = \frac{dh}{dg} \frac{dg}{dx}.\,</math>
 
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===The inverse function rule===
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{{main|inverse functions and differentiation}}
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If the function ''f'' has an [[inverse function]] ''g'', meaning that {{nowrap|1=''g''(''f''(''x'')) = ''x''}} and {{nowrap|1=''f''(''g''(''y'')) = ''y''}}, then
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:<math>g' = \frac{1}{f'\circ g}.\,</math>


In Leibniz notation, this is written as
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==Power laws, polynomials, quotients, and reciprocals==
== Louboutin Malaysia  and sure enough ==
===The polynomial or elementary power rule===


{{main|Power rule}}
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If <math>f(x) = x^n</math>, for any [[integer]] ''n'' then
  <li>[http://ks35439.kimsufi.com/spip.php?article450/ http://ks35439.kimsufi.com/spip.php?article450/]</li>
:<math>f'(x) = nx^{n-1}.\,</math>
 
 
  <li>[http://www.observatoiredesreligions.fr/spip.php?article8 http://www.observatoiredesreligions.fr/spip.php?article8]</li>
Special cases include:
 
* ''Constant rule'': if ''f'' is the constant function ''f''(''x'') = ''c'', for any number ''c'', then for all ''x'', ''f′''(''x'') = 0.
  <li>[http://advertsbook.net/forum/read.php?27,331344 http://advertsbook.net/forum/read.php?27,331344]</li>
* if ''f''(''x'') = ''x'', then ''f′''(''x'') = 1.  This special case may be generalized to:
 
*:''The derivative of an affine function is constant'': if ''f''(''x'') = ''ax'' + ''b'', then ''f′''(''x'') = ''a''.
  <li>[http://enseignement-lsf.com/spip.php?article64#forum24673982 http://enseignement-lsf.com/spip.php?article64#forum24673982]</li>
 
 
Combining this rule with the linearity of the derivative and the addition rule permits the computation of the derivative of any polynomial.
  <li>[http://fatburningsouprecipes.carolasmith.com/forum/profile.php?id=2028 http://fatburningsouprecipes.carolasmith.com/forum/profile.php?id=2028]</li>
 
 
===The reciprocal rule===
</ul>
 
{{main|Reciprocal rule}}
The derivative of ''h''(''x'') = 1/''f''(''x'') for any (nonvanishing) function ''f'' is:
 
:<math> h'(x) = -\frac{f'(x)}{[f(x)]^2}.\ </math>
 
In Leibniz's notation, this is written
:<math> \frac{d(1/f)}{dx} = -\frac{1}{f^2}\frac{df}{dx}.\,</math>
 
The reciprocal rule can be derived from the chain rule and the power rule.
 
===The quotient rule===
 
{{main|Quotient rule}}
 
If ''f'' and ''g'' are functions, then:
:<math>\left(\frac{f}{g}\right)' = \frac{f'g - g'f}{g^2}\quad</math> wherever ''g'' is nonzero.
 
This can be derived from reciprocal rule and the product rule. Conversely (using the constant rule) the reciprocal rule may be derived from the special case ''f''(''x'') = 1.
 
===Generalized power rule===
 
{{main|Power rule}}
 
The elementary power rule generalizes considerably. The most general power rule is the '''functional power rule''': for any functions ''f'' and ''g'',
:<math>(f^g)' = \left(e^{g\ln f}\right)' = f^g\left(f'{g \over f} + g'\ln f\right),\quad</math>
wherever both sides are well defined.
 
Special cases:
* If ''f''(''x'') = ''x''<sup>''a''</sup>, ''f′''(''x'') = ''ax''<sup>''a'' − 1</sup> when ''a'' is any real number and ''x'' is positive.
* The reciprocal rule may be derived as the special case where ''g''(''x'') = −1.
 
== Derivatives of exponential and logarithmic functions ==
 
:<math> \frac{d}{dx}\left(c^{ax}\right) = {c^{ax} \ln c \cdot a } ,\qquad c > 0</math>
note that the equation above is true for all ''c'', but the derivative for c < 0 yields a complex number.
 
:<math> \frac{d}{dx}\left(e^x\right) = e^x</math>
 
:<math> \frac{d}{dx}\left( \log_c x\right) = {1 \over x \ln c} , \qquad c > 0, c \ne 1</math>
the equation above is also true for all ''c'' but yields a complex number if c<0.
 
:<math> \frac{d}{dx}\left( \ln x\right)  = {1 \over x} ,\qquad x \ne 0</math>
 
:<math> \frac{d}{dx}\left( \ln |x|\right) = {|x| \over x^2}</math>
 
:<math> \frac{d}{dx}\left( x^x \right) = x^x(1+\ln x).</math>
 
===Logarithmic derivatives===
 
The [[logarithmic derivative]] is another way of stating the rule for differentiating the [[logarithm]] of a function (using the chain rule):
:<math> (\ln f)'= \frac{f'}{f} \quad</math> wherever ''f'' is positive.
 
== Derivatives of trigonometric functions ==
{{details|Differentiation of trigonometric functions}}
 
{| style="width:100%; background:transparent; margin-left:2em;"
|width=50%|<math> (\sin x)' = \cos x \,</math>
|width=50%|<math> (\arcsin x)' = { 1 \over \sqrt{1 - x^2}} \,</math>
|-
|<math> (\cos x)' = -\sin x \,</math>
|<math> (\arccos x)' = -{1 \over \sqrt{1 - x^2}} \,</math>
|-
|<math> (\tan x)' = \sec^2 x = { 1 \over \cos^2 x} = 1 + \tan^2 x \,</math>
|<math> (\arctan x)' = { 1 \over 1 + x^2} \,</math>
|-
|<math> (\sec x)' = \sec x \tan x \,</math>
|<math> (\operatorname{arcsec} x)' = { 1 \over |x|\sqrt{x^2 - 1}} \,</math>
|-
|<math> (\csc x)' = -\csc x \cot x \,</math>
|<math> (\operatorname{arccsc} x)' = -{1 \over |x|\sqrt{x^2 - 1}} \,</math>
|-
|<math> (\cot x)' = -\csc^2 x = { -1 \over \sin^2 x} = -(1 + \cot^2 x)\,</math>
|<math> (\operatorname{arccot} x)' = -{1 \over 1 + x^2} \,</math>
|}
 
==Derivatives of hyperbolic functions==
{| style="width:100%; background:transparent; margin-left:2em;"
|width=50%|<math>( \sinh x )'= \cosh x = \frac{e^x +
e^{-x}}{2}</math>
|width=50%|<math>(\operatorname{arsinh}\,x)' = { 1 \over \sqrt{x^2 + 1}}</math>
|-
|<math>(\cosh x )'= \sinh x = \frac{e^x - e^{-x}}{2}</math>
|<math>(\operatorname{arcosh}\,x)' = {\frac {1}{\sqrt{x^2-1}}}</math>
|-
|<math>(\tanh x )'= {\operatorname{sech}^2\,x}</math>
|<math>(\operatorname{artanh}\,x)' = { 1 \over 1 - x^2}</math>
|-
|<math>(\operatorname{sech}\,x)' = - \tanh x\,\operatorname{sech}\,x</math>
|<math>(\operatorname{arsech}\,x)' = -{1 \over x\sqrt{1 - x^2}}</math>
|-
|<math>(\operatorname{csch}\,x)' = -\,\operatorname{coth}\,x\,\operatorname{csch}\,x</math>
|<math>(\operatorname{arcsch}\,x)' = -{1 \over |x|\sqrt{1 + x^2}}</math>
|-
|<math>(\operatorname{coth}\,x )' =
 
-\,\operatorname{csch}^2\,x</math>
|<math>(\operatorname{arcoth}\,x)' = -{ 1 \over 1 - x^2}</math>
|}
 
==Derivatives of special functions==
{| style="width:100%; background:transparent; margin-left:2em;"
|width=50%|
;[[Gamma function]]
<math>\Gamma'(x) = \int_0^\infty t^{x-1} e^{-t} \ln t\,dt</math>
:<math>= \Gamma(x) \left(\sum_{n=1}^\infty \left(\ln\left(1 + \dfrac{1}{n}\right) - \dfrac{1}{x + n}\right) - \dfrac{1}{x}\right) = \Gamma(x) \psi(x)</math>
|width=50%|
|}
{| style="width:100%; background:transparent; margin-left:2em;"
|width=50%|
;[[Riemann Zeta function]]
<math>\zeta'(x) = -\sum_{n=1}^\infty \frac{\ln n}{n^x} =
-\frac{\ln 2}{2^x} - \frac{\ln 3}{3^x} - \frac{\ln 4}{4^x} - \cdots
\!</math>
 
:<math>= -\sum_{p \text{ prime}} \frac{p^{-x} \ln p}{(1-p^{-x})^2}\prod_{q \text{ prime}, q \neq p} \frac{1}{1-q^{-x}} \!</math>
|}
 
==Derivatives of integrals==
 
{{main|Differentiation under the integral sign}}
 
Suppose that it is required to differentiate with respect to ''x'' the function
 
:<math>F(x)=\int_{a(x)}^{b(x)}f(x,t)\,dt,</math>
 
where the functions <math>f(x,t)\,</math> and <math>\frac{\partial}{\partial x}\,f(x,t)\,</math> are both continuous in both <math>t\,</math> and <math>x\,</math> in some region of the <math>(t,x)\,</math> plane, including <math>a(x)\leq t\leq b(x),</math> <math>x_0\leq x\leq x_1\,</math>, and the functions <math>a(x)\,</math> and <math>b(x)\,</math> are both continuous and both have continuous derivatives for <math>x_0\leq x\leq x_1\,</math>. Then for <math>\,x_0\leq x\leq x_1\,\,</math>:
 
:<math> F'(x) = f(x,b(x))\,b'(x) - f(x,a(x))\,a'(x) + \int_{a(x)}^{b(x)} \frac{\partial}{\partial x}\, f(x,t)\; dt\,. </math>
 
This formula is the general form of the [[Leibniz integral rule]] and can be derived using the
[[fundamental theorem of calculus]].
 
==Derivatives to ''n''th order==
Some rules exist for computing the ''n''th derivative of functions, where ''n'' is a positive integer.  These include:
 
===Faà di Bruno's formula===
{{main|Faà di Bruno's formula}}
If ''f'' and ''g'' are ''n'' times differentiable, then
 
:<math>  \frac{d^n}{d x^n} [f(g(x))]= n! \sum_{\{k_m\}}^{} f^{(r)}(g(x)) \prod_{m=1}^n \frac{1}{k_m!} \left(g^{(m)}(x) \right)^{k_m}</math>
 
where <math> r = \sum_{m=1}^{n-1} k_m</math> and the set <math> \{k_m\}</math> consists of all non-negative integer solutions of the Diophantine equation <math> \sum_{m=1}^{n} m k_m = n</math>.
 
===General Leibniz rule===
{{main|General Leibniz rule}}
If ''f'' and ''g'' are ''n'' times differentiable, then
 
:<math> \frac{d^n}{dx^n}[f(x)g(x)] = \sum_{k=0}^{n} \binom{n}{k} \frac{d^{n-k}}{d x^{n-k}} f(x) \frac{d^k}{d x^k} g(x)</math>
 
==See also==
 
*[[Derivative]]
*[[Differential calculus]]
*[[Vector calculus identities]]
*[[Differentiable function]]
*[[Differential of a function]]
*[[Limit of a function]]
*[[Function (mathematics)]]
*[[List of mathematical functions]]
*[[Trigonometric functions]]
*[[Inverse trigonometric functions]]
*[[Hyperbolic functions]]
*[[Inverse hyperbolic functions]]
*[[Matrix calculus]]
*[[Differentiation under the integral sign]]
 
==References==
{{reflist}}
 
==Sources and further reading==
These rules are given in many books, both on elementary and advanced calculus, in pure and applied mathematics. Those in this article (in addition to the above references) can be found in:
*''Mathematical Handbook of Formulas and Tables (3rd edition)'', S. Lipschutz, M.R. Spiegel, J. Liu, Schuam's Outline Series, 2009, ISBN 978-0-07-154855-7.
*''The Cambridge Handbook of Physics Formulas'', G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2.
*''Mathematical methods for physics and engineering'', K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, ISBN 978-0-521-86153-3
*''NIST Handbook of Mathematical Functions'', F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark, Cambridge University Press, 2010, ISBN 978-0-521-19225-5.
 
==External links==
{{Library resources box
|by=no
|onlinebooks=no
|others=no
|about=yes
|label=Differentiation rules}}
 
* [http://www.planetcalc.com/675/ Derivative calculator with formula simplification]
* [http://mathmajor.org/calculus-and-analysis/table-of-derivatives/ A Table of Derivatives]
 
[[Category:Differential calculus|*]]
[[Category:Differentiation rules]]
[[Category:Mathematics-related lists|Derivatives]]
[[Category:Mathematical tables|Derivatives]]
[[Category:Mathematical identities]]
 
[[ar:قائمة المطابقات التفاضلية]]
[[bs:Tablica izvoda]]
[[ca:Taula de derivades]]
[[es:Tabla de derivadas]]
[[fr:Dérivées usuelles]]
[[he:נגזרת]]
[[pl:Pochodna_funkcji#Pochodne_funkcji_elementarnych]]
[[sq:Tabela e derivateve]]
[[sl:tabela odvodov]]

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