18 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Differentiation rules | 2/3 | https://en.wikipedia.org/wiki/Differentiation_rules | reference | science, encyclopedia | 2026-05-05T08:13:51.299386+00:00 | kb-cron |
h
′
(
x
)
=
−
f
′
(
x
)
(
f
(
x
)
)
2
,
{\displaystyle h'(x)=-{\frac {f'(x)}{(f(x))^{2}}},}
wherever
f
{\textstyle f}
is nonzero. In Leibniz's notation, this formula is written:
d
(
1
f
)
d
x
=
−
1
f
2
d
f
d
x
.
{\displaystyle {\frac {d\left({\frac {1}{f}}\right)}{dx}}=-{\frac {1}{f^{2}}}{\frac {df}{dx}}.}
The reciprocal rule can be derived either from the quotient rule or from the combination of power rule and chain rule.
=== Quotient rule ===
If
f
{\textstyle f}
and
g
{\textstyle 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}}},}
wherever
g
{\textstyle g}
is nonzero. This can be derived from the product rule and the reciprocal rule.
=== Generalized power rule ===
The elementary power rule generalizes considerably. The most general power rule is the functional power rule: for any functions
f
{\textstyle f}
and
g
{\textstyle 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
{\textstyle f(x)=x^{a}}
, then
f
′
(
x
)
=
a
x
a
−
1
{\textstyle f'(x)=ax^{a-1}}
when
a
{\textstyle a}
is any nonzero real number and
x
{\textstyle x}
is positive. The reciprocal rule may be derived as the special case where
g
(
x
)
=
−
1
{\textstyle g(x)=-1\!}
.
== Derivatives of exponential and logarithmic functions ==
d
d
x
(
c
a
x
)
=
a
c
a
x
ln
c
,
c
>
0.
{\displaystyle {\frac {d}{dx}}\left(c^{ax}\right)={ac^{ax}\ln c},\qquad c>0.}
The equation above is true for all
c
{\displaystyle c}
, but the derivative for
c
<
0
{\displaystyle 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
>
1.
{\displaystyle {\frac {d}{dx}}\left(\log _{c}x\right)={1 \over x\ln c},\qquad c>1.}
The equation above is also true for all
c
{\textstyle c}
but yields a complex number if
c
<
0
{\textstyle 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
,
x
≠
0.
{\displaystyle {\frac {d}{dx}}\left(\ln |x|\right)={1 \over x},\qquad x\neq 0.}
d
d
x
(
W
(
x
)
)
=
1
x
+
e
W
(
x
)
,
x
>
−
1
e
,
{\displaystyle {\frac {d}{dx}}\left(W(x)\right)={1 \over {x+e^{W(x)}}},\qquad x>-{1 \over e},}
where
W
(
x
)
{\textstyle W(x)}
is the Lambert W function.
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
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
d
f
i
d
x
exists.
{\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},\qquad {\text{ if }}f_{i<n}(x)>0{\text{ and }}{\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}},}
wherever
f
{\textstyle f}
is positive. Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative. 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 ==
The derivatives in the table above are for when the range of the inverse secant is
[
0
,
π
]
{\textstyle [0,\pi ]}
and when the range of the inverse cosecant is
[
−
π
2
,
π
2
]
{\textstyle \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]}
. It is common to additionally define an inverse tangent function with two arguments,
arctan
(
y
,
x
)
{\textstyle \arctan(y,x)}
. Its value lies in the range
[
−
π
,
π
]
{\textstyle [-\pi ,\pi ]}
and reflects the quadrant of the point
(
x
,
y
)
{\textstyle (x,y)}
. For the first and fourth quadrant (i.e.,
x
>
0
{\displaystyle x>0}
), one has
arctan
(
y
,
x
>
0
)
=
arctan
(
y
x
)
{\textstyle \arctan(y,x>0)=\arctan({\frac {y}{x}})}
. Its partial derivatives are:
∂
arctan
(
y
,
x
)
∂
y
=
x
x
2
+
y
2
and
∂
arctan
(
y
,
x
)
∂
x
=
−
y
x
2
+
y
2
.
{\displaystyle {\frac {\partial \arctan(y,x)}{\partial y}}={\frac {x}{x^{2}+y^{2}}}\qquad {\text{and}}\qquad {\frac {\partial \arctan(y,x)}{\partial x}}={\frac {-y}{x^{2}+y^{2}}}.}
== Derivatives of hyperbolic functions ==
== Derivatives of special functions ==
=== Gamma function ===
Γ
(
x
)
=
∫
0
∞
t
x
−
1
e
−
t
d
t
{\displaystyle \Gamma (x)=\int _{0}^{\infty }t^{x-1}e^{-t}\,dt}