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---
title: "Differentiation rules"
chunk: 3/3
source: "https://en.wikipedia.org/wiki/Differentiation_rules"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T08:13:51.299386+00:00"
instance: "kb-cron"
---
Γ
′
(
x
)
=
∫
0
∞
t
x
−
1
e
−
t
ln
⁡
t
d
t
=
Γ
(
x
)
(
∑
n
=
1
∞
(
ln
⁡
(
1
+
1
n
)
−
1
x
+
n
)
−
1
x
)
=
Γ
(
x
)
ψ
(
x
)
,
{\displaystyle {\begin{aligned}\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{aligned}}}
with
ψ
(
x
)
{\textstyle \psi (x)}
being the digamma function, expressed by the parenthesized expression to the right of
Γ
(
x
)
{\textstyle \Gamma (x)}
in the line above.
=== Riemann zeta function ===
ζ
(
x
)
=
∑
n
=
1
∞
1
n
x
{\displaystyle \zeta (x)=\sum _{n=1}^{\infty }{\frac {1}{n^{x}}}}
ζ
′
(
x
)
=
−
∑
n
=
1
∞
ln
⁡
n
n
x
=
−
ln
⁡
2
2
x
−
ln
⁡
3
3
x
−
ln
⁡
4
4
x
−
⋯
=
−
∑
p
prime
p
−
x
ln
⁡
p
(
1
−
p
−
x
)
2
∏
q
prime
,
q
≠
p
1
1
−
q
−
x
{\displaystyle {\begin{aligned}\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{aligned}}}
== Derivatives of integrals ==
Suppose that it is required to differentiate with respect to
x
{\textstyle 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
)
{\textstyle f(x,t)}
and
∂
∂
x
f
(
x
,
t
)
{\textstyle {\frac {\partial }{\partial x}}\,f(x,t)}
are both continuous in both
t
{\textstyle t}
and
x
{\textstyle x}
in some region of the
(
t
,
x
)
{\textstyle (t,x)}
plane, including
a
(
x
)
≤
t
≤
b
(
x
)
{\textstyle a(x)\leq t\leq b(x)}
, where
x
0
≤
x
≤
x
1
{\textstyle x_{0}\leq x\leq x_{1}}
, and the functions
a
(
x
)
{\textstyle a(x)}
and
b
(
x
)
{\textstyle b(x)}
are both continuous and both have continuous derivatives for
x
0
≤
x
≤
x
1
{\textstyle x_{0}\leq x\leq x_{1}}
. Then, for
x
0
≤
x
≤
x
1
{\textstyle \,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 nth order ==
Some rules exist for computing the
n
{\textstyle n}
th derivative of functions, where
n
{\textstyle n}
is a positive integer, including:
=== Faà di Bruno's formula ===
If
f
{\textstyle f}
and
g
{\textstyle g}
are
n
{\textstyle 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
{\textstyle r=\sum _{m=1}^{n-1}k_{m}}
and the set
{
k
m
}
{\textstyle \{k_{m}\}}
consists of all non-negative integer solutions of the Diophantine equation
∑
m
=
1
n
m
k
m
=
n
{\textstyle \sum _{m=1}^{n}mk_{m}=n}
.
=== General Leibniz rule ===
If
f
{\textstyle f}
and
g
{\textstyle g}
are
n
{\textstyle 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 ==
Differentiable function – Mathematical function whose derivative exists
Differential of a function – Notion in calculus
Differentiation of integrals – Problem of the derivative of the mean value integral
Differentiation under the integral sign – Differentiation under the integral sign formulaPages displaying short descriptions of redirect targets
Hyperbolic functions – Hyperbolic analogues of trigonometric functions
Inverse hyperbolic functions – Mathematical functions
Inverse trigonometric functions – Inverse functions of sin, cos, tan, etc.
Lists of integrals
List of mathematical functions
Matrix calculus – Specialized notation for multivariable calculus
Trigonometric functions – Functions of an angle
Vector calculus identities – Mathematical identities
== References ==
== 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 ==
Derivative calculator with formula simplification
The table of derivatives with animated proves
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