View source for Differentiation rules

{{Short description|Rules for computing derivatives of functions}}
{{Calculus |Differential}}

This is a summary of '''differentiation rules''', that is, rules for computing the [[derivative]] of a [[function (mathematics)|function]] in [[calculus]].

== Elementary rules of differentiation ==

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, Schaum's Outline Series, 2009, {{ISBN|978-0-07-150861-2}}.</ref><ref>''Advanced Calculus (3rd edition)'', R. Wrede, M.R. Spiegel, Schaum's Outline Series, 2010, {{ISBN|978-0-07-162366-7}}.</ref> — including the case of [[complex number|complex numbers ('''C''')]].<ref>''Complex Variables'', M.R. Spiegel, S. Lipschutz, J.J. Schiller, D. Spellman, Schaum's Outlines Series, McGraw Hill (USA), 2009, {{ISBN|978-0-07-161569-3}}</ref>

===Constant term rule===
For any value of <math>c</math>, where <math>c \in \mathbb{R}</math>, if <math>f(x)</math> is the constant function given by <math>f(x) = c</math>, then <math>\frac{df}{dx} = 0</math>.<ref>{{cite web |title=Differentiation Rules |url=https://courseware.cemc.uwaterloo.ca/11/assignments/47/6 |website=University of Waterloo - CEMC Open Courseware |access-date=3 May 2022}}</ref>

====Proof====
Let <math>c \in \mathbb{R}</math> and <math>f(x) = c</math>. By the definition of the derivative,

:<math>\begin{align}
f'(x) &= \lim_{h \to 0}\frac{f(x + h) - f(x)}{h} \\
&= \lim_{h \to 0} \frac{(c) - (c)}{h} \\
&= \lim_{h \to 0} \frac{0}{h} \\
&= \lim_{h \to 0} 0 \\
&= 0
\end{align}</math>

This shows that the derivative of any constant function is 0.

===Differentiation is linear===
{{main|Linearity of differentiation}}

For any functions <math>f</math> and <math>g</math> and any real numbers <math>a</math> and <math>b</math>, the derivative of the function <math>h(x) = af(x) + bg(x)</math> with respect to <math>x</math> is: <math> h'(x) = a f'(x) + b g'(x).</math>

In [[Leibniz's notation]] this is written as:
<math display="block"> \frac{d(af+bg)}{dx}  = a\frac{df}{dx} +b\frac{dg}{dx}.</math>

Special cases include:
* The ''constant factor rule'' <math display="block">(af)' = af' </math>
* The ''sum rule'' <math display="block">(f + g)' = f' + g'</math>
* The ''subtraction rule'' <math display="block">(f - g)' = f' - g'.</math>

===The product rule===

{{main|Product rule}}

For the functions ''f'' and ''g'', the derivative of the function ''h''(''x'') = ''f''(''x'') ''g''(''x'') with respect to ''x'' is
<math display="block"> h'(x) = (fg)'(x) = f'(x) g(x) + f(x) g'(x).</math>
In Leibniz's notation this is written
<math display="block">\frac{d(fg)}{dx} = \frac{df}{dx} g + f \frac{dg}{dx}.</math>

===The chain rule===
{{main|Chain rule}}

The derivative of the function <math>h(x) = f(g(x))</math> is
<math display="block"> h'(x) = f'(g(x))\cdot g'(x).</math>

In Leibniz's notation, this is written as:
<math display="block">\frac{d}{dx}h(x) = \left.\frac{d}{dz}f(z)\right|_{z=g(x)}\cdot \frac{d}{dx}g(x),</math>
often abridged to
<math display="block">\frac{dh(x)}{dx} = \frac{df(g(x))}{dg(x)} \cdot \frac{dg(x)}{dx}.</math>

Focusing on the notion of maps, and the differential being a map <math>\text{D}</math>, this is written in a more concise way as:
<math display="block"> [\text{D} (f\circ g)]_x = [\text{D} f]_{g(x)} \cdot [\text{D}g]_x\,.</math>

===The inverse function rule===

{{main|Inverse functions and differentiation}}

If the function {{Mvar|f}} has an [[inverse function]] {{Mvar|g}}, meaning that <math>g(f(x)) = x</math> and <math>f(g(y)) = y,</math> then
<math display="block">g' = \frac{1}{f'\circ g}.</math>

In Leibniz notation, this is written as
<math display="block"> \frac{dx}{dy} = \frac{1}{\frac{dy}{dx}}.</math>

==Power laws, polynomials, quotients, and reciprocals==
===The polynomial or elementary power rule===

{{main|Power rule}}

If <math>f(x) = x^r</math>, for any real number <math>r \neq 0,</math> then 
:<math>f'(x) = rx^{r-1}.</math>

When <math>r = 1,</math> this becomes the special case that if <math>f(x) = x,</math> then <math>f'(x) = 1.</math>

Combining the power rule with the sum and constant multiple rules permits the computation of the derivative of any polynomial.

===The reciprocal rule===

{{main|Reciprocal rule}}
The derivative of <math>h(x)=\frac{1}{f(x)}</math>for any (nonvanishing) function ''{{Mvar|f}}'' is:

:<math> h'(x) = -\frac{f'(x)}{(f(x))^2}</math> wherever ''{{Mvar|f}}'' is non-zero.

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 either from the quotient rule, or from the combination of power rule and chain rule.

===The quotient rule===
{{main|Quotient rule}}
If ''{{Mvar|f}}'' and ''{{Mvar|g}}'' are functions, then:
:<math>\left(\frac{f}{g}\right)' = \frac{f'g - g'f}{g^2}\quad</math> wherever ''{{Mvar|g}}'' is nonzero.

This can be derived from the product rule and the reciprocal rule.

===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 ''{{Mvar|f}}'' and ''{{Mvar|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 <math display="inline">f(x)=x^a\!</math>, then <math display="inline">f'(x)=ax^{a-1}</math>when ''{{Mvar|a}}'' is any non-zero real number and ''{{Mvar|x}}'' is positive.
* The reciprocal rule may be derived as the special case where <math display="inline">g(x)=-1\!</math>.

== Derivatives of exponential and logarithmic functions ==

:<math> \frac{d}{dx}\left(c^{ax}\right) = {ac^{ax} \ln c } ,\qquad c > 0</math>
the equation above is true for all {{Mvar|c}}, but the derivative for <math display="inline">c<0</math> yields a complex number.

:<math> \frac{d}{dx}\left(e^{ax}\right) = ae^{ax}</math>

:<math> \frac{d}{dx}\left( \log_c x\right) = {1 \over x \ln c} , \qquad c > 1</math>

the equation above is also true for all ''{{Mvar|c}}'', but yields a complex number if <math display="inline">c<0\!</math>.

:<math> \frac{d}{dx}\left( \ln x\right)  = {1 \over x} ,\qquad x > 0.</math>

:<math> \frac{d}{dx}\left( \ln |x|\right) = {1 \over x} ,\qquad x \neq 0.</math>

:<math> \frac{d}{dx}\left( W(x)\right) = {1 \over {x+e^{W(x)}}} ,\qquad x > -{1 \over e}.\qquad</math>where <math>W(x)</math> is the [[Lambert W function]]

:<math> \frac{d}{dx}\left( x^x \right) = x^x(1+\ln x).</math>

:<math> \frac{d}{dx}\left( f(x)^{ g(x) } \right ) = g(x)f(x)^{g(x)-1} \frac{df}{dx} + f(x)^{g(x)}\ln{( f(x) )}\frac{dg}{dx}, \qquad \text{if }f(x) > 0, \text{ and if } \frac{df}{dx} \text{ and } \frac{dg}{dx} \text{ exist.}</math>

:<math> \frac{d}{dx}\left( f_{1}(x)^{f_{2}(x)^{\left ( ... \right )^{f_{n}(x)}}} \right ) = \left [\sum\limits_{k=1}^{n} \frac{\partial }{\partial x_{k}} \left( f_{1}(x_1)^{f_{2}(x_2)^{\left ( ... \right )^{f_{n}(x_n)}}} \right ) \right ] \biggr\vert_{x_1 = x_2 = ... =x_n = x}, \text{ if } f_{i<n}(x) > 0 \text{ and }</math> <math> \frac{df_{i}}{dx} \text{ exists. }</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 ''{{Mvar|f}}'' is positive.

[[Logarithmic differentiation]] is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative.{{citation needed|date=October 2021}}

Logarithms can be used to remove exponents, convert products into sums, and convert division into subtraction — each of which may lead to a simplified expression for taking derivatives.

== Derivatives of trigonometric functions ==
{{main|Differentiation of trigonometric functions}}

{| style="width:100%; background:transparent; margin-left:2em;"
|width=50%|<math> (\sin x)' = \cos x = \frac{e^{ix} +
 e^{-ix}}{2} </math>
|width=50%|<math> (\arcsin x)' = { 1 \over \sqrt{1 - x^2}} </math>
|-
|<math> (\cos x)' = -\sin x = \frac{e^{-ix} -
 e^{ix}}{2i} </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> (\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>
|-
|<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>
|}
The derivatives in the table above are for when the range of the inverse secant is <math>[0,\pi]\!</math> and when the range of the inverse cosecant is <math>\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\!</math>.

It is common to additionally define an [[Atan2|inverse tangent function with two arguments]], <math>\arctan(y,x)\!</math>.  Its value lies in the range <math>[-\pi,\pi]\!</math> and reflects the quadrant of the point <math>(x,y)\!</math>.  For the first and fourth quadrant (i.e. <math>x > 0\!</math>) one has <math>\arctan(y, x>0) = \arctan(y/x)\!</math>.  Its partial derivatives are
{| style="width:100%; background:transparent; margin-left:2em;"
| width="100%" |<math> \frac{\partial \arctan(y,x)}{\partial y} = \frac{x}{x^2 + y^2}</math>, and <math> \frac{\partial \arctan(y,x)}{\partial x} = \frac{-y}{x^2 + y^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{arcsinh}x)' = { 1 \over \sqrt{1 + x^2}}</math>
|-
|<math>(\cosh x )'= \sinh x = \frac{e^x - e^{-x}}{2}</math>
|<math>(\operatorname{arccosh}x)' = {\frac {1}{\sqrt{x^2-1}}}</math>
|-
|<math>(\tanh x )'= {\operatorname{sech}^2x} = { 1 \over \cosh^2 x} = 1 - \tanh^2 x</math>
|<math>(\operatorname{arctanh}x)' = { 1 \over 1 - x^2}</math>
|-
|<math>(\coth x )' = -\operatorname{csch}^2x = -{ 1 \over \sinh^2 x} = 1 - \coth^2 x</math>
|<math>(\operatorname{arccoth}x)' = { 1 \over 1 - x^2}</math>
|-
|<math>(\operatorname{sech} x)' = -\operatorname{sech}{x}\tanh{x}</math>
|<math>(\operatorname{arcsech}x)' = -{1 \over x\sqrt{1 - x^2}}</math>
|-
|<math>(\operatorname{csch}x)' = -\operatorname{csch}{x}\coth{x}</math>
|<math>(\operatorname{arccsch}x)' = -{1 \over |x|\sqrt{1 + x^2}}</math>
|}
See [[Hyperbolic functions#Derivatives|Hyperbolic functions]] for restrictions on these derivatives.

==Derivatives of special functions==
;[[Gamma function]]
:<math>\Gamma(x) = \int_0^\infty t^{x-1} e^{-t}\, dt</math>
:<math>\begin{align}
\Gamma'(x) & = \int_0^\infty t^{x-1} e^{-t} \ln t\,dt \\
& = \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)
\end{align}</math> {{pb}} with <math>\psi(x)</math> being the [[digamma function]], expressed by the parenthesized expression to the right of <math>\Gamma(x)</math> in the line above.
;[[Riemann Zeta function]]
:<math>\zeta(x) = \sum_{n=1}^\infty \frac{1}{n^x}</math>
:<math>\begin{align}
\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 \\
& = -\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}}
\end{align}</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 {{mvar|n}}-th derivative of functions, where {{mvar|n}} is a positive integer.  These include:

===Faà di Bruno's formula===
{{main|Faà di Bruno's formula}}
If {{mvar|f}} and {{mvar|g}} are {{mvar|n}}-times differentiable, then
<math display="block">  \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 display="inline"> 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 display="inline"> \sum_{m=1}^{n} m k_m = n</math>.

===General Leibniz rule===
{{main|General Leibniz rule}}
If {{mvar|f}} and {{mvar|g}} are {{mvar|n}}-times differentiable, then
<math display="block"> \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==

* {{annotated link|Differentiable function}}
* {{annotated link|Differential of a function}}
* {{annotated link|Differentiation of integrals}}
* {{annotated link|Differentiation under the integral sign}}
* {{annotated link|Hyperbolic functions}}
* {{annotated link|Inverse hyperbolic functions}}
* {{annotated link|Inverse trigonometric functions}}
* {{annotated link|Lists of integrals}}
* {{annotated link|List of mathematical functions}}
* {{annotated link|Matrix calculus}}
* {{annotated link|Trigonometric functions}}
* {{annotated link|Vector calculus identities}}

==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, Schaum'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 
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* [http://www.planetcalc.com/675/ Derivative calculator with formula simplification]

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