Functional power

Power rules

''This article concerns power rules for computing the derivative in calculus

In mathematics, the power rule is a method for differentiating expressions involving exponentiation (the power operation). The most basic such rule is the elementary power rule which states that the derivative of the function f(x) = xn, where n is a natural number, is f '(x) = n xn−1. In other words, in Leibniz's notation,

frac{d}{dx} x^n = nx^{n-1}.
See Calculus with polynomials for further elaboration of this rule and its applications.

The elementary power rule generalizes very easily to the case that n is replaced by any real power p, where xp, for x positive, is defined to be exp(p ln(x) ). Here "exp" denotes the exponential function from the real numbers to the positive real numbers, and ln denotes the natural logarithm, which is the inverse function of exp. The exponential function is commonly denoted exp(x) = ex, where e (≈ 2.718) is the base of the natural logarithm, with ln(e) = 1.

The power rule with real exponent, and other power rules, follow easily from the formulae for the derivatives of the exponential and the (natural) logarithm:

  • exp'(x) = exp(x)
  • ln'(x) = 1/x.

There are various ways to prove these formulae, depending on how the exponential and logarithm are defined. Since they are mutually inverse functions, the two formulae are related by the inverse function rule for differentiation. They can both be established from power series definitions; alternatively, the second formula is immediate from the fundamental theorem of calculus and the definition of the logarithm in terms of the area under the graph of the function 1/x.

The most general power rule is the functional power rule: for any functions f and g,

(f^g)' = f^gleft(g'ln f + {gf' over f} right),quad
wherever both sides are well defined.

This follows by writing fg as exp(g ln (f) ). Then by the chain rule and the formula for exp', the derivative is

(f^g)' = e^left(gln fright)(gln f)'
The product rule applied to the second term, together with the chain rule and the formula for ln' give
(f^g)' = f^gleft(g'ln f + {gf' over f} right),quad
as required.

The power rule for the function xp follows immediately, since in this case g is constant, and f ' = 1.

Another important special case is the function bx, with b constant, whose derivative is bx ln b.

y = x^a

e^y = e^left(x^aright)

e^yleft(frac{dy}{dx}right) = ax^left(a-1right)e^left(x^aright)

e^y = e^left(x^aright)

e^left(x^aright)left(frac{dy}{dx}right) = ax^left(a-1right)e^left(x^aright)

frac{dy}{dx} = ax^left(a-1right)

See also

Differentiation rules

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