The power rule is one of the basic rules of differential calculus in mathematics . It is used to determine the derivation of power functions .
Definition and scope
The derivative of the function is . This applies to and .
f
(
x
)
=
x
n
{\ displaystyle f (x) = x ^ {n}}
f
′
(
x
)
=
n
⋅
x
n
-
1
{\ displaystyle f '(x) = n \ cdot x ^ {n-1}}
n
∈
{
2
,
3
,
4th
,
...
}
{\ displaystyle n \ in \ {2,3,4, \ ldots {} \}}
x
∈
R.
{\ displaystyle x \ in \ mathbb {R}}
For example, the function has the derivative .
f
(
x
)
=
x
4th
{\ displaystyle f (x) = x ^ {4}}
f
′
(
x
)
=
4th
⋅
x
3
{\ displaystyle f '(x) = 4 \ cdot x ^ {3}}
The power rule applies to only for , since the expression would appear at position 0 , the definition of which is not unique.
n
=
1
{\ displaystyle n = 1}
x
∈
R.
∖
{
0
}
{\ displaystyle x \ in \ mathbb {R} \ setminus \ {0 \}}
0
0
{\ displaystyle 0 ^ {0}}
The power rule applies to only for , otherwise a division by 0 would occur.
n
∈
{
-
1
,
-
2
,
-
3
,
...
}
{\ displaystyle n \ in \ {- 1, -2, -3, \ ldots {} \}}
x
∈
R.
∖
{
0
}
{\ displaystyle x \ in \ mathbb {R} \ setminus \ {0 \}}
The power rule also applies to power functions if the exponent ( exponent ) is not an integer, but only in the range :
f
(
x
)
=
x
s
{\ displaystyle f (x) = x ^ {s}}
s
∈
R.
{\ displaystyle s \ in \ mathbb {R}}
x
∈
R.
,
x
>
0
{\ displaystyle x \ in \ mathbb {R}, x> 0}
f
′
(
x
)
=
s
⋅
x
s
-
1
{\ displaystyle f '(x) = s \ cdot x ^ {s-1}}
Derivation
1st case: The exponent is a natural number
The derivation of a power function at the point x is the limit value :
(
x
n
)
′
=
lim
Δ
x
→
0
(
x
+
Δ
x
)
n
-
x
n
Δ
x
{\ displaystyle (x ^ {n}) '= \ lim _ {\ Delta x \ to 0} {\ frac {(x + \ Delta x) ^ {n} -x ^ {n}} {\ Delta x}} }
.
According to the binomial theorem , this is the same
lim
Δ
x
→
0
(
n
0
)
x
n
+
(
n
1
)
x
n
-
1
Δ
x
+
(
n
2
)
x
n
-
2
Δ
x
2
+
⋯
+
(
n
n
-
1
)
x
Δ
x
n
-
1
+
(
n
n
)
Δ
x
n
-
x
n
Δ
x
{\ displaystyle \ lim _ {\ Delta x \ to 0} {\ frac {{n \ choose 0} x ^ {n} + {n \ choose 1} x ^ {n-1} {\ Delta x} + { n \ choose 2} x ^ {n-2} {\ Delta x} ^ {2} + \ dots + {n \ choose n-1} x {\ Delta x} ^ {n-1} + {n \ choose n} {\ Delta x} ^ {n} -x ^ {n}} {\ Delta x}}}
written with so-called binomial coefficients . The power rule follows from this:
(
x
n
)
′
=
lim
Δ
x
→
0
(
n
1
)
x
n
-
1
Δ
x
+
(
n
2
)
x
n
-
2
Δ
x
2
+
⋯
+
(
n
n
-
1
)
x
Δ
x
n
-
1
+
(
n
n
)
Δ
x
n
Δ
x
{\ displaystyle (x ^ {n}) '= \ lim _ {\ Delta x \ to 0} {\ frac {{n \ choose 1} x ^ {n-1} {\ Delta x} + {n \ choose 2} x ^ {n-2} {\ Delta x} ^ {2} + \ dots + {n \ choose n-1} x {\ Delta x} ^ {n-1} + {n \ choose n} { \ Delta x} ^ {n}} {\ Delta x}}}
=
lim
Δ
x
→
0
[
(
n
1
)
x
n
-
1
+
(
n
2
)
x
n
-
2
Δ
x
+
⋯
+
(
n
n
-
1
)
x
Δ
x
n
-
2
+
(
n
n
)
Δ
x
n
-
1
]
{\ displaystyle = \ lim _ {\ Delta x \ to 0} \ left [{n \ choose 1} x ^ {n-1} + {n \ choose 2} x ^ {n-2} {\ Delta x} + \ dots + {n \ choose n-1} x {\ Delta x} ^ {n-2} + {n \ choose n} {\ Delta x} ^ {n-1} \ right]}
=
(
n
1
)
x
n
-
1
=
n
⋅
x
n
-
1
{\ displaystyle = {n \ choose 1} x ^ {n-1} = n \ cdot x ^ {n-1}}
.
Illustrated in the figure, an 'n-dimensional cube' grows in exactly n directions (along the n coordinate axes) by '(n-1) -dimensional cubes'. A square grows (or crystallizes) marginally by 2 side lines, and a cube grows by 3 squares.
2nd case: any exponent
One uses the representation with the help of the exponential function :
and derives with the help of the chain rule and the derivation rule for the exponential function:
x
s
=
(
e
ln
x
)
s
=
e
s
⋅
ln
x
{\ displaystyle x ^ {s} = (e ^ {\ ln x}) ^ {s} = e ^ {s \ cdot \ ln x}}
(
x
s
)
′
=
(
e
s
⋅
ln
x
)
′
=
e
s
⋅
ln
x
⋅
(
s
⋅
ln
x
)
′
{\ displaystyle (x ^ {s}) '= (e ^ {s \ cdot \ ln x})' = e ^ {s \ cdot \ ln x} \ cdot (s \ cdot \ ln x) '}
The factor rule and the rule for the derivation of the logarithm function are used for the inner derivative :
(
s
⋅
ln
x
)
′
=
s
⋅
1
x
{\ displaystyle (s \ cdot \ ln x) '= s \ cdot {\ frac {1} {x}}}
By putting this in and writing for again , you get
(
e
s
⋅
ln
x
)
{\ displaystyle (e ^ {s \ cdot \ ln x})}
x
s
{\ displaystyle x ^ {s}}
(
x
s
)
′
=
x
s
⋅
s
⋅
1
x
=
s
⋅
x
s
-
1
{\ displaystyle (x ^ {s}) '= x ^ {s} \ cdot s \ cdot {\ frac {1} {x}} = s \ cdot x ^ {s-1}}
This derivation only applies to . For, however , the function can also be differentiated at the point and the rule also applies to the point . One calculates directly using the difference quotient:
x
≠
0
{\ displaystyle x \ neq 0}
s
>
1
{\ displaystyle s> 1}
f
(
x
)
=
x
s
{\ displaystyle f (x) = x ^ {s}}
x
=
0
{\ displaystyle x = 0}
x
=
0
{\ displaystyle x = 0}
f
′
(
0
)
=
lim
Δ
x
→
0
(
Δ
x
)
s
-
0
s
Δ
x
-
0
=
lim
Δ
x
→
0
(
Δ
x
)
s
-
1
=
0
=
s
⋅
0
s
-
1
{\ displaystyle f '(0) = \ lim _ {\ Delta x \ to 0} {\ frac {(\ Delta x) ^ {s} -0 ^ {s}} {\ Delta x-0}} = \ lim _ {\ Delta x \ to 0} (\ Delta x) ^ {s-1} = 0 = s \ cdot 0 ^ {s-1}}
Multiple derivative of a power function with natural exponent
(For the notation of the following see Leibniz notation .) Within the domain of definition of a power function with a natural exponent , its -fold derivative is ...
n
{\ displaystyle n}
k
{\ displaystyle k}
... for .
1
≤
k
≤
n
:
d
k
d
x
k
x
n
=
n
!
(
n
-
k
)
!
⋅
x
n
-
k
{\ displaystyle 1 \ leq k \ leq n: \ quad {\ frac {d ^ {k}} {dx ^ {k}}} x ^ {n} = {\ frac {n!} {(nk)!} } \ cdot x ^ {nk}}
proof
The claim can be proved for with complete induction .
k
≤
n
{\ displaystyle k \ leq n}
Induction start for (true)
k
=
1
:
d
d
x
x
n
=
n
⋅
x
n
-
1
=
n
!
(
n
-
1
)
!
x
n
-
1
{\ displaystyle k = 1: \ quad {\ frac {d} {dx}} x ^ {n} = n \ cdot x ^ {n-1} = {\ frac {n!} {(n-1)! }} x ^ {n-1}}
Induction requirement:
d
k
d
x
k
x
n
=
n
!
(
n
-
k
)
!
⋅
x
n
-
k
{\ displaystyle \ quad \ quad {\ frac {d ^ {k}} {dx ^ {k}}} x ^ {n} = {\ frac {n!} {(nk)!}} \ cdot x ^ { nk}}
Induction claim:
d
k
+
1
d
x
k
+
1
x
n
=
n
!
(
n
-
k
-
1
)
!
⋅
x
n
-
k
-
1
{\ displaystyle \ quad \ quad \ quad {\ frac {d ^ {k + 1}} {dx ^ {k + 1}}} x ^ {n} = {\ frac {n!} {(nk-1) !}} \ cdot x ^ {nk-1}}
Induction step:
d
k
+
1
d
x
k
+
1
x
n
=
{\ displaystyle \ quad {\ frac {d ^ {k + 1}} {dx ^ {k + 1}}} x ^ {n} =}
The -th derivative is the derivative of the -th derivative:
(
k
+
1
)
{\ displaystyle (k + 1)}
k
{\ displaystyle k}
d
d
x
d
k
d
x
k
x
n
=
{\ displaystyle \ quad {\ frac {d} {dx}} {\ frac {d ^ {k}} {dx ^ {k}}} x ^ {n} =}
with the induction assumption:
d
d
x
n
!
(
n
-
k
)
!
⋅
x
n
-
k
=
{\ displaystyle \ quad {\ frac {d} {dx}} {\ frac {n!} {(nk)!}} \ cdot x ^ {nk} =}
n
!
(
n
-
k
)
!
⋅
(
n
-
k
)
⋅
x
n
-
k
-
1
=
{\ displaystyle \ quad {\ frac {n!} {(nk)!}} \ cdot (nk) \ cdot x ^ {nk-1} =}
n
!
(
n
-
k
-
1
)
!
⋅
x
n
-
k
-
1
{\ displaystyle \ quad {\ frac {n!} {(nk-1)!}} \ cdot x ^ {nk-1}}
, q. e. d.
For some applications it is useful to define a function as the -th derivative of itself. As can be easily seen, the rule for also applies .
0
{\ displaystyle 0}
k
=
0
{\ displaystyle k = 0}
For is in particular
k
=
n
{\ displaystyle k = n}
d
n
d
x
n
x
n
=
n
!
{\ displaystyle {\ frac {d ^ {n}} {dx ^ {n}}} x ^ {n} = n!}
...For
k
>
n
:
d
k
d
x
k
x
n
=
0
{\ displaystyle k> n: \ quad {\ frac {d ^ {k}} {dx ^ {k}}} x ^ {n} = 0}
This follows directly from the fact that the derivative of any constant function is the null function ; the latter also applies to the null function itself.
d
n
d
x
n
x
n
=
n
!
{\ displaystyle {\ frac {d ^ {n}} {dx ^ {n}}} x ^ {n} = n!}
Individual evidence
↑ Otto Forster : Analysis 1. Differential and integral calculus of a variable. Vieweg-Verlag (1976), ISBN 3-499-27024-2 , §15, example (15.9)
↑ Otto Forster : Analysis 1. Differential and integral calculus of a variable. Vieweg-Verlag (1976), ISBN 3-499-27024-2 , §15, example (15.16)
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">