15 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Differentiation rules | 1/3 | https://en.wikipedia.org/wiki/Differentiation_rules | reference | science, encyclopedia | 2026-05-05T08:13:51.299386+00:00 | kb-cron |
This article 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
{\textstyle \mathbb {R} }
) that return real values, although, more generally, the formulas below apply wherever they are well defined, including the case of complex numbers (
C
{\textstyle \mathbb {C} }
).
=== Constant term rule === For any value of
c
{\textstyle c}
, where
c
∈
R
{\textstyle c\in \mathbb {R} }
, if
f
(
x
)
{\textstyle f(x)}
is the constant function given by
f
(
x
)
=
c
{\textstyle f(x)=c}
, then
d
f
d
x
=
0
{\textstyle {\frac {df}{dx}}=0}
.
==== Proof ==== Let
c
∈
R
{\textstyle c\in \mathbb {R} }
and
f
(
x
)
=
c
{\textstyle f(x)=c}
. By the definition of the derivative:
f
′
(
x
)
=
lim
h
→
0
f
(
x
+
h
)
−
f
(
x
)
h
=
lim
h
→
0
(
c
)
−
(
c
)
h
=
lim
h
→
0
0
h
=
lim
h
→
0
0
=
0.
{\displaystyle {\begin{aligned}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{aligned}}}
This computation shows that the derivative of any constant function is 0.
==== Intuitive (geometric) explanation ==== The derivative of the function at a point is the slope of the line tangent to the curve at the point. The slope of the constant function is 0, because the tangent line to the constant function is horizontal and its angle is 0. In other words, the value of the constant function,
y
{\textstyle y}
, will not change as the value of
x
{\textstyle x}
increases or decreases.
=== Linearity of differentiation ===
For any functions
f
{\textstyle f}
and
g
{\textstyle g}
and any real numbers
a
{\textstyle a}
and
b
{\textstyle b}
, the derivative of the function
h
(
x
)
=
a
f
(
x
)
+
b
g
(
x
)
{\textstyle h(x)=af(x)+bg(x)}
with respect to
x
{\textstyle x}
is
h
′
(
x
)
=
a
f
′
(
x
)
+
b
g
′
(
x
)
{\textstyle h'(x)=af'(x)+bg'(x)}
. In Leibniz's notation, this formula 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:
The constant factor rule:
(
a
f
)
′
=
a
f
′
,
{\displaystyle (af)'=af',}
The sum rule:
(
f
+
g
)
′
=
f
′
+
g
′
,
{\displaystyle (f+g)'=f'+g',}
The difference rule:
(
f
−
g
)
′
=
f
′
−
g
′
.
{\displaystyle (f-g)'=f'-g'.}
=== Product rule ===
For the functions
f
{\textstyle f}
and
g
{\textstyle g}
, the derivative of the function
h
(
x
)
=
f
(
x
)
g
(
x
)
{\textstyle h(x)=f(x)g(x)}
with respect to
x
{\textstyle x}
is:
h
′
(
x
)
=
(
f
g
)
′
(
x
)
=
f
′
(
x
)
g
(
x
)
+
f
(
x
)
g
′
(
x
)
.
{\displaystyle h'(x)=(fg)'(x)=f'(x)g(x)+f(x)g'(x).}
In Leibniz's notation, this formula is written:
d
(
f
g
)
d
x
=
g
d
f
d
x
+
f
d
g
d
x
.
{\displaystyle {\frac {d(fg)}{dx}}=g{\frac {df}{dx}}+f{\frac {dg}{dx}}.}
=== Chain rule ===
The derivative of the function
h
(
x
)
=
f
(
g
(
x
)
)
{\textstyle h(x)=f(g(x))}
is:
h
′
(
x
)
=
f
′
(
g
(
x
)
)
⋅
g
′
(
x
)
.
{\displaystyle h'(x)=f'(g(x))\cdot g'(x).}
In Leibniz's notation, this formula is written as:
d
d
x
h
(
x
)
=
d
d
z
f
(
z
)
|
z
=
g
(
x
)
⋅
d
d
x
g
(
x
)
,
{\displaystyle {\frac {d}{dx}}h(x)=\left.{\frac {d}{dz}}f(z)\right|_{z=g(x)}\cdot {\frac {d}{dx}}g(x),}
often abridged to:
d
h
(
x
)
d
x
=
d
f
(
g
(
x
)
)
d
g
(
x
)
⋅
d
g
(
x
)
d
x
.
{\displaystyle {\frac {dh(x)}{dx}}={\frac {df(g(x))}{dg(x)}}\cdot {\frac {dg(x)}{dx}}.}
Focusing on the notion of maps, and the differential being a map
D
{\textstyle {\text{D}}}
, this formula is written in a more concise way as:
[
D
(
f
∘
g
)
]
x
=
[
D
f
]
g
(
x
)
⋅
[
D
g
]
x
.
{\displaystyle [{\text{D}}(f\circ g)]_{x}=[{\text{D}}f]_{g(x)}\cdot [{\text{D}}g]_{x}.}
=== Inverse function rule ===
If the function
f
{\textstyle f}
has an inverse function
g
{\textstyle g}
, meaning that
g
(
f
(
x
)
)
=
x
{\textstyle g(f(x))=x}
and
f
(
g
(
y
)
)
=
y
{\textstyle f(g(y))=y}
, then:
g
′
=
1
f
′
∘
g
.
{\displaystyle g'={\frac {1}{f'\circ g}}.}
In Leibniz notation, this formula is written as:
d
x
d
y
=
1
d
y
d
x
.
{\displaystyle {\frac {dx}{dy}}={\frac {1}{\frac {dy}{dx}}}.}
== Power laws, polynomials, quotients, and reciprocals ==
=== Polynomial or elementary power rule ===
If
f
(
x
)
=
x
r
{\textstyle f(x)=x^{r}}
, for any real number
r
≠
0
{\textstyle r\neq 0}
, then:
f
′
(
x
)
=
r
x
r
−
1
.
{\displaystyle f'(x)=rx^{r-1}.}
When
r
=
1
{\textstyle r=1}
, this formula becomes the special case that, if
f
(
x
)
=
x
{\textstyle f(x)=x}
, then
f
′
(
x
)
=
1
{\textstyle f'(x)=1}
. Combining the power rule with the sum and constant multiple rules permits the computation of the derivative of any polynomial.
=== Reciprocal rule ===
The derivative of
h
(
x
)
=
1
f
(
x
)
{\textstyle h(x)={\frac {1}{f(x)}}}
for any (nonvanishing) function
f
{\textstyle f}
is: