Power rule

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In calculus, the power rule is one of the most important differentiation rules. Since differentiation is linear, polynomials can be differentiated using this rule.

${\frac {d}{dx}}x^{n}=nx^{{n-1}},\qquad n\neq 0.$

The power rule holds for all powers except for the constant value $x^{0}$ which is covered by the constant rule. The derivative is just $0$ rather than $0\cdot x^{{-1}}$ which is undefined when $x=0$ .

The inverse of the power rule enables all powers of a variable $x$ except $x^{{-1}}$ to be integrated. This integral is called Cavalieri's quadrature formula and was first found in a geometric form by Bonaventura Cavalieri for $n\geq 0$ . It is considered the first general theorem of calculus to be discovered.

$\int \!x^{n}\,dx={\frac {x^{{n+1}}}{n+1}}+C,\qquad n\neq -1.$

This is an indefinite integral where $C$ is the arbitrary constant of integration.

The integration of $x^{{-1}}$ requires a separate rule.

$\int \!x^{{-1}}\,dx=\ln |x|+C,$

Hence, the derivative of $x^{{100}}$ is $100x^{{99}}$ and the integral of $x^{{100}}$ is ${\frac {1}{101}}x^{{101}}+C$ .

Power rule

Historically the power rule was derived as the inverse of Cavalieri's quadrature formula which gave the area under $x^{n}$ for any integer $n\geq 0$ . Nowadays the power rule is derived first and integration considered as its inverse.

For integers $n\geq 1$ , the derivative of $f(x)=x^{n}\!$ is $f'(x)=nx^{{n-1}},\!$ that is,

$\left(x^{n}\right)'=nx^{{n-1}}.$

The power rule for integration

$\int \!x^{n}\,dx={\frac {x^{{n+1}}}{n+1}}+C$

for $n\geq 0$ is then an easy consequence. One just needs to take the derivative of this equality and use the power rule and linearity of differentiation on the right-hand side.

Proof

To prove the power rule for differentiation, we use the definition of the derivative as a limit. But first, note the factorization for $n\geq 1$ :

$f(x)-f(a)=x^{n}-a^{n}=(x-a)(x^{{n-1}}+ax^{{n-2}}+\cdots +a^{{n-2}}x+a^{{n-1}})$

Using this, we can see that

$f'(a)=\lim _{{x\rightarrow a}}{\frac {x^{n}-a^{n}}{x-a}}=\lim _{{x\rightarrow a}}x^{{n-1}}+ax^{{n-2}}+\cdots +a^{{n-2}}x+a^{{n-1}}$

Since the division has been eliminated and we have a continuous function, we can freely substitute to find the limit:

$f'(a)=\lim _{{x\rightarrow a}}x^{{n-1}}+ax^{{n-2}}+\cdots +a^{{n-2}}x+a^{{n-1}}=a^{{n-1}}+a^{{n-1}}+\cdots +a^{{n-1}}+a^{{n-1}}=n\cdot a^{{n-1}}$

The use of the quotient rule allows the extension of this rule for n as a negative integer, and the use of the laws of exponents and the chain rule allows this rule to be extended to all rational values of $n$ . For an irrational $n$ , a rational approximation is appropriate.

Differentiation of arbitrary polynomials

To differentiate arbitrary polynomials, one can use the linearity property of the differential operator to obtain:

$\left(\sum _{{r=0}}^{n}a_{r}x^{r}\right)'=\sum _{{r=0}}^{n}\left(a_{r}x^{r}\right)'=\sum _{{r=0}}^{n}a_{r}\left(x^{r}\right)'=\sum _{{r=0}}^{n}ra_{r}x^{{r-1}}.$

Using the linearity of integration and the power rule for integration, one shows in the same way that

$\int \!\left(\sum _{{k=0}}^{n}a_{k}x^{k}\right)\,dx=\sum _{{k=0}}^{n}{\frac {a_{k}x^{{k+1}}}{k+1}}+C.$

Generalizations

One can prove that the power rule is valid for any exponent $r$ , that is

$\left(x^{r}\right)'=rx^{{r-1}},$

as long as $x$ is in the domain of the functions on the left and right hand sides and $r$ is nonzero. Using this formula, together with

$\int \!x^{{-1}}\,dx=\ln |x|+C,$

one can differentiate and integrate linear combinations of powers of $x$ which are not necessarily polynomials.

References

Larson, Ron; Hostetler, Robert P.; and Edwards, Bruce H. (2003). Calculus of a Single Variable: Early Transcendental Functions (3rd edition). Houghton Mifflin Company. ISBN 0-618-22307-X.

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