This is a summary of differentiation rules , that is, rules for computing the derivative of a function in calculus .
Elementary rules of differentiation
Unless otherwise stated, all functions are functions of real numbers (R ) that return real values; although more generally, the formulae below apply wherever they are well defined [ 1] [ 2] complex numbers (C ) .[ 3]
Differentiation is linear
For any functions f and g and any real numbers a and b the derivative of the function h (x ) = af (x ) + bg (x )x is
h
′
(
x
)
=
a
f
′
(
x
)
+
b
g
′
(
x
)
.
{\displaystyle h'(x)=af'(x)+bg'(x).\,}
In Leibniz's notation this is written as:
d
(
a
f
+
b
g
)
d
x
=
a
d
f
d
x
+
b
d
g
d
x
.
{\displaystyle {\frac {d(af+bg)}{dx}}=a{\frac {df}{dx}}+b{\frac {dg}{dx}}.}
Special cases include:
(
a
f
)
′
=
a
f
′
{\displaystyle (af)'=af'\,}
(
f
+
g
)
′
=
f
′
+
g
′
{\displaystyle (f+g)'=f'+g'\,}
(
f
−
g
)
′
=
f
′
−
g
′
.
{\displaystyle (f-g)'=f'-g'.\,}
The product rule
For the functions f and g , the derivative of the function h (x ) = f (x ) g (x )
with respect to x is
h
′
(
x
)
=
f
′
(
x
)
g
(
x
)
+
f
(
x
)
g
′
(
x
)
.
{\displaystyle h'(x)=f'(x)g(x)+f(x)g'(x).\,}
In Leibniz's notation this is written
d
(
f
g
)
d
x
=
d
f
d
x
g
+
f
d
g
d
x
.
{\displaystyle {\frac {d(fg)}{dx}}={\frac {df}{dx}}g+f{\frac {dg}{dx}}.}
The chain rule
The derivative of the function of a function h (x ) = f (g (x )) with respect to x is
h
′
(
x
)
=
f
′
(
g
(
x
)
)
g
′
(
x
)
.
{\displaystyle h'(x)=f'(g(x))g'(x).\,}
In Leibniz's notation this is written as:
d
h
d
x
=
d
f
(
g
(
x
)
)
d
g
(
x
)
d
g
(
x
)
d
x
.
{\displaystyle {\frac {dh}{dx}}={\frac {df(g(x))}{dg(x)}}{\frac {dg(x)}{dx}}.\,}
However, by relaxing the interpretation of h as a function, this is often simply written
d
h
d
x
=
d
h
d
g
d
g
d
x
.
{\displaystyle {\frac {dh}{dx}}={\frac {dh}{dg}}{\frac {dg}{dx}}.\,}
The inverse function rule
If the function f has an inverse function g , meaning that g (f (x )) = x f (g (y )) = y
g
′
=
1
f
′
∘
g
.
{\displaystyle g'={\frac {1}{f'\circ g}}.}
In Leibniz notation, this is written as
d
x
d
y
=
1
d
y
/
d
x
.
{\displaystyle {\frac {dx}{dy}}={\frac {1}{dy/dx}}.}
The polynomial or elementary power rule
If
f
(
x
)
=
x
n
{\displaystyle f(x)=x^{n}}
n
≠
0
{\displaystyle n\neq 0}
f
′
(
x
)
=
n
x
n
−
1
.
{\displaystyle f'(x)=nx^{n-1}.\,}
Special cases include:
Constant rule : if
n
=
0
{\displaystyle n=0}
f is the constant function f (x ) = c , for some number c , and, for all x , f′ (x ) = 0.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 . Combining this rule with the linearity of the derivative and the addition rule permits the computation of the derivative of any polynomial.
The reciprocal rule
The derivative of h (x ) = 1/f (x ) for any (nonvanishing) function f is:
h
′
(
x
)
=
−
f
′
(
x
)
(
f
(
x
)
)
2
.
{\displaystyle h'(x)=-{\frac {f'(x)}{(f(x))^{2}}}.\ }
In Leibniz's notation, this is written
d
(
1
/
f
)
d
x
=
−
1
f
2
d
f
d
x
.
{\displaystyle {\frac {d(1/f)}{dx}}=-{\frac {1}{f^{2}}}{\frac {df}{dx}}.\,}
The reciprocal rule can be derived from the chain rule and the power rule.
The quotient rule
If f and g are functions, then:
(
f
g
)
′
=
f
′
g
−
g
′
f
g
2
{\displaystyle \left({\frac {f}{g}}\right)'={\frac {f'g-g'f}{g^{2}}}\quad }
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
The elementary power rule generalizes considerably. The most general power rule is the functional power rule : for any functions f and g ,
(
f
g
)
′
=
(
e
g
ln
f
)
′
=
f
g
(
f
′
g
f
+
g
′
ln
f
)
,
{\displaystyle (f^{g})'=\left(e^{g\ln f}\right)'=f^{g}\left(f'{g \over f}+g'\ln f\right),\quad }
wherever both sides are well defined.
Special cases:
If f (x ) = x a f′ (x ) = ax a − 1a 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
d
d
x
(
c
a
x
)
=
c
a
x
ln
c
⋅
a
,
c
>
0
{\displaystyle {\frac {d}{dx}}\left(c^{ax}\right)={c^{ax}\ln c\cdot a},\qquad c>0}
note that the equation above is true for all c , but the derivative for c < 0 yields a complex number.
d
d
x
(
e
a
x
)
=
a
e
a
x
{\displaystyle {\frac {d}{dx}}\left(e^{ax}\right)=ae^{ax}}
d
d
x
(
log
c
x
)
=
1
x
ln
c
,
c
>
0
,
c
≠
1
{\displaystyle {\frac {d}{dx}}\left(\log _{c}x\right)={1 \over x\ln c},\qquad c>0,c\neq 1}
the equation above is also true for all c but yields a complex number if c<0.
d
d
x
(
ln
x
)
=
1
x
,
x
>
0.
{\displaystyle {\frac {d}{dx}}\left(\ln x\right)={1 \over x},\qquad x>0.}
d
d
x
(
ln
|
x
|
)
=
1
x
.
{\displaystyle {\frac {d}{dx}}\left(\ln |x|\right)={1 \over x}.}
d
d
x
(
x
x
)
=
x
x
(
1
+
ln
x
)
.
{\displaystyle {\frac {d}{dx}}\left(x^{x}\right)=x^{x}(1+\ln x).}
d
d
x
(
f
(
x
)
g
(
x
)
)
=
g
(
x
)
f
(
x
)
g
(
x
)
−
1
d
f
d
x
+
f
(
x
)
g
(
x
)
ln
(
f
(
x
)
)
d
g
d
x
,
if
f
(
x
)
>
0
,
and if
d
f
d
x
and
d
g
d
x
exist.
{\displaystyle {\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.}}}
d
d
x
(
f
1
(
x
)
f
2
(
x
)
(
.
.
.
)
f
n
(
x
)
)
=
[
∑
k
=
1
n
∂
∂
x
k
(
f
1
(
x
1
)
f
2
(
x
2
)
(
.
.
.
)
f
n
(
x
n
)
)
]
|
x
1
=
x
2
=
.
.
.
=
x
n
=
x
,
if
f
i
<
n
(
x
)
>
0
and
{\displaystyle {\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 }}}
d
f
i
d
x
exists.
{\displaystyle {\frac {df_{i}}{dx}}{\text{ exists. }}}
Logarithmic derivatives
The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule):
(
ln
f
)
′
=
f
′
f
{\displaystyle (\ln f)'={\frac {f'}{f}}\quad }
f is positive.
Derivatives of trigonometric functions
(
sin
x
)
′
=
cos
x
{\displaystyle (\sin x)'=\cos x\,}
(
arcsin
x
)
′
=
1
1
−
x
2
{\displaystyle (\arcsin x)'={1 \over {\sqrt {1-x^{2}}}}\,}
(
cos
x
)
′
=
−
sin
x
{\displaystyle (\cos x)'=-\sin x\,}
(
arccos
x
)
′
=
−
1
1
−
x
2
{\displaystyle (\arccos x)'=-{1 \over {\sqrt {1-x^{2}}}}\,}
(
tan
x
)
′
=
sec
2
x
=
1
cos
2
x
=
1
+
tan
2
x
{\displaystyle (\tan x)'=\sec ^{2}x={1 \over \cos ^{2}x}=1+\tan ^{2}x\,}
(
arctan
x
)
′
=
1
1
+
x
2
{\displaystyle (\arctan x)'={1 \over 1+x^{2}}\,}
(
sec
x
)
′
=
sec
x
tan
x
{\displaystyle (\sec x)'=\sec x\tan x\,}
(
arcsec
x
)
′
=
1
|
x
|
x
2
−
1
{\displaystyle (\operatorname {arcsec} x)'={1 \over |x|{\sqrt {x^{2}-1}}}\,}
(
csc
x
)
′
=
−
csc
x
cot
x
{\displaystyle (\csc x)'=-\csc x\cot x\,}
(
arccsc
x
)
′
=
−
1
|
x
|
x
2
−
1
{\displaystyle (\operatorname {arccsc} x)'=-{1 \over |x|{\sqrt {x^{2}-1}}}\,}
(
cot
x
)
′
=
−
csc
2
x
=
−
1
sin
2
x
=
−
(
1
+
cot
2
x
)
{\displaystyle (\cot x)'=-\csc ^{2}x={-1 \over \sin ^{2}x}=-(1+\cot ^{2}x)\,}
(
arccot
x
)
′
=
−
1
1
+
x
2
{\displaystyle (\operatorname {arccot} x)'=-{1 \over 1+x^{2}}\,}
It is common to additionally define an inverse tangent function with two arguments,
arctan
(
y
,
x
)
{\displaystyle \arctan(y,x)}
[
−
π
,
π
]
{\displaystyle [-\pi ,\pi ]}
(
x
,
y
)
{\displaystyle (x,y)}
x
>
0
{\displaystyle x>0}
arctan
(
y
,
x
>
0
)
=
arctan
(
y
/
x
)
{\displaystyle \arctan(y,x>0)=\arctan(y/x)}
∂
arctan
(
y
,
x
)
∂
y
=
x
x
2
+
y
2
{\displaystyle {\frac {\partial \arctan(y,x)}{\partial y}}={\frac {x}{x^{2}+y^{2}}}}
∂
arctan
(
y
,
x
)
∂
x
=
−
y
x
2
+
y
2
.
{\displaystyle {\frac {\partial \arctan(y,x)}{\partial x}}={\frac {-y}{x^{2}+y^{2}}}.}
Derivatives of hyperbolic functions
(
sinh
x
)
′
=
cosh
x
=
e
x
+
e
−
x
2
{\displaystyle (\sinh x)'=\cosh x={\frac {e^{x}+e^{-x}}{2}}}
(
arsinh
x
)
′
=
1
x
2
+
1
{\displaystyle (\operatorname {arsinh} \,x)'={1 \over {\sqrt {x^{2}+1}}}}
(
cosh
x
)
′
=
sinh
x
=
e
x
−
e
−
x
2
{\displaystyle (\cosh x)'=\sinh x={\frac {e^{x}-e^{-x}}{2}}}
(
arcosh
x
)
′
=
1
x
2
−
1
{\displaystyle (\operatorname {arcosh} \,x)'={\frac {1}{\sqrt {x^{2}-1}}}}
(
tanh
x
)
′
=
sech
2
x
{\displaystyle (\tanh x)'={\operatorname {sech} ^{2}\,x}}
(
artanh
x
)
′
=
1
1
−
x
2
{\displaystyle (\operatorname {artanh} \,x)'={1 \over 1-x^{2}}}
(
sech
x
)
′
=
−
tanh
x
sech
x
{\displaystyle (\operatorname {sech} \,x)'=-\tanh x\,\operatorname {sech} \,x}
(
arsech
x
)
′
=
−
1
x
1
−
x
2
{\displaystyle (\operatorname {arsech} \,x)'=-{1 \over x{\sqrt {1-x^{2}}}}}
(
csch
x
)
′
=
−
coth
x
csch
x
{\displaystyle (\operatorname {csch} \,x)'=-\,\operatorname {coth} \,x\,\operatorname {csch} \,x}
(
arcsch
x
)
′
=
−
1
|
x
|
1
+
x
2
{\displaystyle (\operatorname {arcsch} \,x)'=-{1 \over |x|{\sqrt {1+x^{2}}}}}
(
coth
x
)
′
=
−
csch
2
x
{\displaystyle (\operatorname {coth} \,x)'=-\,\operatorname {csch} ^{2}\,x}
(
arcoth
x
)
′
=
1
1
−
x
2
{\displaystyle (\operatorname {arcoth} \,x)'={1 \over 1-x^{2}}}
Derivatives of special functions
Gamma function
Γ
′
(
x
)
=
∫
0
∞
t
x
−
1
e
−
t
ln
t
d
t
{\displaystyle \Gamma '(x)=\int _{0}^{\infty }t^{x-1}e^{-t}\ln t\,dt}
=
Γ
(
x
)
(
∑
n
=
1
∞
(
ln
(
1
+
1
n
)
−
1
x
+
n
)
−
1
x
)
=
Γ
(
x
)
ψ
(
x
)
{\displaystyle =\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)}
Riemann Zeta function
ζ
′
(
x
)
=
−
∑
n
=
1
∞
ln
n
n
x
=
−
ln
2
2
x
−
ln
3
3
x
−
ln
4
4
x
−
⋯
{\displaystyle \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 \!}
=
−
∑
p
prime
p
−
x
ln
p
(
1
−
p
−
x
)
2
∏
q
prime
,
q
≠
p
1
1
−
q
−
x
{\displaystyle =-\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}}}\!}
Derivatives of integrals
Suppose that it is required to differentiate with respect to x the function
F
(
x
)
=
∫
a
(
x
)
b
(
x
)
f
(
x
,
t
)
d
t
,
{\displaystyle F(x)=\int _{a(x)}^{b(x)}f(x,t)\,dt,}
where the functions
f
(
x
,
t
)
{\displaystyle f(x,t)\,}
∂
∂
x
f
(
x
,
t
)
{\displaystyle {\frac {\partial }{\partial x}}\,f(x,t)\,}
t
{\displaystyle t\,}
x
{\displaystyle x\,}
(
t
,
x
)
{\displaystyle (t,x)\,}
a
(
x
)
≤
t
≤
b
(
x
)
,
{\displaystyle a(x)\leq t\leq b(x),}
x
0
≤
x
≤
x
1
{\displaystyle x_{0}\leq x\leq x_{1}\,}
a
(
x
)
{\displaystyle a(x)\,}
b
(
x
)
{\displaystyle b(x)\,}
x
0
≤
x
≤
x
1
{\displaystyle x_{0}\leq x\leq x_{1}\,}
x
0
≤
x
≤
x
1
{\displaystyle \,x_{0}\leq x\leq x_{1}\,\,}
F
′
(
x
)
=
f
(
x
,
b
(
x
)
)
b
′
(
x
)
−
f
(
x
,
a
(
x
)
)
a
′
(
x
)
+
∫
a
(
x
)
b
(
x
)
∂
∂
x
f
(
x
,
t
)
d
t
.
{\displaystyle 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\,.}
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:
If f and g are n times differentiable, then
d
n
d
x
n
[
f
(
g
(
x
)
)
]
=
n
!
∑
{
k
m
}
f
(
r
)
(
g
(
x
)
)
∏
m
=
1
n
1
k
m
!
(
g
(
m
)
(
x
)
)
k
m
{\displaystyle {\frac {d^{n}}{dx^{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}}}
where
r
=
∑
m
=
1
n
−
1
k
m
{\displaystyle r=\sum _{m=1}^{n-1}k_{m}}
{
k
m
}
{\displaystyle \{k_{m}\}}
∑
m
=
1
n
m
k
m
=
n
{\displaystyle \sum _{m=1}^{n}mk_{m}=n}
General Leibniz rule
If f and g are n times differentiable, then
d
n
d
x
n
[
f
(
x
)
g
(
x
)
]
=
∑
k
=
0
n
(
n
k
)
d
n
−
k
d
x
n
−
k
f
(
x
)
d
k
d
x
k
g
(
x
)
{\displaystyle {\frac {d^{n}}{dx^{n}}}[f(x)g(x)]=\sum _{k=0}^{n}{\binom {n}{k}}{\frac {d^{n-k}}{dx^{n-k}}}f(x){\frac {d^{k}}{dx^{k}}}g(x)}
See also
References
^ Calculus (5th edition) , F. Ayres, E. Mendelson, Schuam's Outline Series, 2009, ISBN 978-0-07-150861-2.^ Advanced Calculus (3rd edition) , R. Wrede, M.R. Spiegel, Schuam's Outline Series, 2010, ISBN 978-0-07-162366-7.^ 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
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 978-0-521-86153-3NIST 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
![{\displaystyle {\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 }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e52623821c09e999ec428d4b2264ed2123e6c5c)
![{\displaystyle [-\pi ,\pi ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb064fd6c55820cfa660eabeeda0f6e3c4935ae6)
![{\displaystyle {\frac {d^{n}}{dx^{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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6db9fc6a48b3658dc330625f3c027300228b82b0)
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