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Power rule

The power rule is a fundamental principle in calculus for computing the derivative of a power function, stating that if f(x) = x^n where n is any real number, then f'(x) = n x^{n-1}.^[1] This rule provides a quick method to differentiate expressions involving exponents without resorting to the limit definition of the derivative in every case.^[2] It applies broadly to integer, rational, and irrational exponents, though for negative or fractional powers, the function is typically defined for x > 0 to avoid issues with complex numbers or undefined values at zero.^[1] The power rule emerged during the early development of calculus in the 17th century, with Pierre de Fermat deriving it as part of his methods for finding maxima, minima, and tangents to curves around 1630–1650. Building on precursors like Fermat's method of adequality and Isaac Barrow's method of tangents, the rule was formalized within the broader framework of infinitesimal calculus independently invented by Isaac Newton and Gottfried Wilhelm Leibniz in the 1660s and 1670s.^[3] Newton approached it through fluxions and series expansions, while Leibniz used differentials, but both recognized its utility for polynomial differentiation.^[3] As a cornerstone of differential calculus, the power rule enables efficient computation of derivatives for polynomials and is essential for applications in physics and engineering, where rates of change are modeled. It extends to more advanced forms through the chain rule for composite functions and combines with linearity for sums and scalar multiples, forming the basis for differentiating a wide array of algebraic expressions.^[2] Proofs of the rule typically rely on the binomial theorem for positive integers, algebraic manipulation for negatives, and limits or logarithmic differentiation for general reals.^[1]

Statement

Differentiation

The power rule for differentiation states that if f(x) = x^r where r is a real number, then the derivative is f'(x) = r x^{r-1}.^[4] This rule applies to power functions of the form x^r, enabling the computation of derivatives for a wide range of exponents.^[5] The rule holds under the condition that x lies in the domain where x^r is defined and differentiable, typically x > 0 for non-integer r to ensure the function is real-valued and smooth. For the special case r = 0, f(x) = x^0 = 1 is a constant function, so its derivative is f'(x) = 0.^[6] To illustrate, consider f(x) = x^3; applying the rule yields f'(x) = 3x^{2}.^[5] Similarly, for f(x) = x^{1/2} = \sqrt{x}, the derivative is f'(x) = \frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}}.^[4] The intuition for the rule in the case of positive integer exponents arises from the limit definition of the derivative, where \lim_{h \to 0} \frac{(x+h)^n - x^n}{h} simplifies algebraically to n x^{n-1}.^[5] This power rule serves as the counterpart to the integration power rule, which finds antiderivatives of similar forms.^[6]

Integration

The power rule for integration, often referred to as the reverse power rule, states that for a real number r \neq -1 and x > 0,

\int x^r \, dx = \frac{x^{r+1}}{r+1} + C,

where C is the constant of integration.^[7] This formula applies to indefinite integrals of power functions and serves as the inverse operation to the power rule for differentiation.^[7] A special case arises when r = -1, where the power rule does not apply directly due to division by zero; instead,

\int x^{-1} \, dx = \int \frac{1}{x} \, dx = \ln |x| + C.

This logarithmic form is derived from the limit definition of the integral and is fundamental for integrating rational functions with negative exponents.^[7] For example, applying the general power rule to \int x^2 \, dx yields \frac{x^3}{3} + C, as the exponent increases from 2 to 3 and is divided by 3.^[7] Similarly, for a fractional exponent like \int x^{-1/2} \, dx, the result is $2x^{1/2} + C, where the exponent becomes $1/2 and division by $1/2 multiplies by 2.^[7] To verify these antiderivatives, differentiation of the integrated form returns the original integrand: for instance, the derivative of \frac{x^{r+1}}{r+1} + C is x^r, confirming the rule's consistency with the differentiation power rule.^[7]

Proofs

Integer Exponents

The power rule for differentiation states that if f(x) = x^n, where n is a positive integer, then f'(x) = n x^{n-1}.^[4] This result can be established rigorously using mathematical induction on n.^[8] To prove the rule by induction, begin with the base case where n = 1. For f(x) = x^1 = x, the derivative is f'(x) = 1, which matches $1 \cdot x^{1-1} = 1 \cdot x^0 = 1, using the known derivative of x.^[4] Assume the statement holds for some positive integer k, so if g(x) = x^k, then g'(x) = k x^{k-1}.^[8] For the inductive step, consider n = k + 1, so f(x) = x^{k+1} = x \cdot x^k. Differentiating using the product rule gives:

f'(x) = (1) \cdot x^k + x \cdot (k x^{k-1}) = x^k + k x^k = (k + 1) x^k,

which equals (k + 1) x^{(k+1)-1}.^[4] By the principle of mathematical induction, the power rule holds for all positive integers n.^[8] An alternative proof for positive integers uses the definition of the derivative and the binomial theorem. Consider f(x) = x^n, so

f'(x) = \lim_{h \to 0} \frac{(x + h)^n - x^n}{h}.

Expanding (x + h)^n via the binomial theorem yields

(x + h)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} h^k = x^n + n x^{n-1} h + \sum_{k=2}^n \binom{n}{k} x^{n-k} h^k.

Substituting into the limit expression gives

f'(x) = \lim_{h \to 0} \left( n x^{n-1} + \sum_{k=2}^n \binom{n}{k} x^{n-k} h^{k-1} \right).

As h \to 0, the sum vanishes, leaving f'(x) = n x^{n-1}.^[5] The power rule extends to negative integers by rewriting x^{-m} = 1 / x^m for positive integer m. Let f(x) = x^{-m}, so f(x) = 1 / u(x) where u(x) = x^m and u'(x) = m x^{m-1}. Applying the quotient rule (with numerator constant 1, so its derivative is 0) yields

f'(x) = \frac{0 \cdot x^m - 1 \cdot m x^{m-1}}{(x^m)^2} = \frac{-m x^{m-1}}{x^{2m}} = -m x^{m-1 - 2m} = -m x^{-m-1}.

Since n = -m, this simplifies to n x^{n-1}.^[9] Thus, the rule holds for all integers n \neq 0.^[9]

Rational Exponents

The power rule for differentiation extends to rational exponents, allowing the derivative of x^r where r = p/q and p, q are integers with q \neq 0, to be computed as r x^{r-1}, assuming x > 0 for even q to ensure real values. This generalization relies on previously established rules for integer exponents and tools like the chain rule or implicit differentiation.^[10] One standard proof uses the chain rule by rewriting the expression. Consider x^{p/q} = (x^p)^{1/q}. Let u = x^p, so the function is u^{1/q}. The derivative is then \frac{d}{dx} [u^{1/q}] = \frac{1}{q} u^{1/q - 1} \cdot \frac{du}{dx}. Since \frac{du}{dx} = p x^{p-1} by the integer power rule, substitute to get \frac{1}{q} (x^p)^{1/q - 1} \cdot p x^{p-1} = \frac{p}{q} (x^p)^{1/q - 1} x^{p-1}. Simplifying the exponent, (x^p)^{1/q - 1} = x^{p(1/q - 1)} = x^{p/q - p}, so the expression becomes \frac{p}{q} x^{p/q - p} x^{p-1} = \frac{p}{q} x^{p/q - p + p - 1} = \frac{p}{q} x^{p/q - 1}. Thus, \frac{d}{dx} x^{p/q} = \frac{p}{q} x^{p/q - 1}.^[11] An alternative proof employs implicit differentiation. Let y = x^{p/q}, so y^q = x^p. Differentiate both sides with respect to x: the left side gives q y^{q-1} \frac{dy}{dx} by the chain rule and integer power rule, while the right side is p x^{p-1}. Solving for \frac{dy}{dx}, we have \frac{dy}{dx} = \frac{p x^{p-1}}{q y^{q-1}}. Substitute y = x^{p/q} to obtain \frac{dy}{dx} = \frac{p x^{p-1}}{q (x^{p/q})^{q-1}} = \frac{p x^{p-1}}{q x^{p - p/q}} = \frac{p}{q} x^{p-1 - (p - p/q)} = \frac{p}{q} x^{p/q - 1}. This confirms the rule for rational exponents.^[12]^[13]

Real Exponents

The power rule for real exponents states that if f(x) = x^r where r is any real number, then f'(x) = r x^{r-1} for x > 0. This form completes the generalization of the power rule beyond integers and rationals, allowing differentiation of functions like x^{\sqrt{2}} or x^{\pi}. The result holds under the assumption that x^r is defined as e^{r \ln x}, ensuring a real-valued function on the positive reals.^[14] One standard proof uses logarithmic differentiation. Let y = x^r, so \ln y = r \ln x. Differentiating both sides with respect to x gives \frac{1}{y} \frac{dy}{dx} = \frac{r}{x}, using the derivative of \ln x. Solving for the derivative yields \frac{dy}{dx} = y \cdot \frac{r}{x} = x^r \cdot \frac{r}{x} = r x^{r-1}. This confirms the rule, assuming the differentiability of the natural logarithm on (0, \infty).^[14] An alternative proof applies the chain rule directly to the exponential form. Let f(x) = e^{r \ln x}. The derivative is f'(x) = e^{r \ln x} \cdot \frac{d}{dx} (r \ln x) = x^r \cdot r \cdot \frac{1}{x} = r x^{r-1}, using the derivative of the exponential function and the natural logarithm.^[14] For the special case r = 0, f(x) = x^0 = 1 (a constant for x > 0), so f'(x) = 0, which matches $0 \cdot x^{-1} = 0. The domain restriction to x > 0 ensures \ln x is defined and x^r remains real-valued; for non-integer r, negative bases generally yield complex values. The result for rational exponents serves as a dense approximation to real exponents via continuity of the power function.^[14]

Historical Development

Early Discoveries

The early developments of the power rule emerged in the 1630s through Bonaventura Cavalieri's method of indivisibles, which treated geometric figures as aggregates of infinitely thin lines to compute areas under power curves. Cavalieri applied this technique to determine the areas bounded by curves corresponding to positive integer powers, such as the parabola, by comparing sums of these indivisibles between figures of equal height, establishing ratios that prefigured integration rules for such functions. His approach, detailed in works like the Exercitationes geometricae sex (1647), relied on superposition of lines to equate areas without algebraic manipulation, marking an intuitive step toward handling power functions geometrically.^[15]^[16] Concurrently, Pierre de Fermat advanced the differentiation aspect of power functions in the 1630s by developing a geometric method of tangents to find maxima and minima. Fermat's technique involved constructing secant lines and using "adequation"—an approximate equality that effectively eliminated higher-order terms—to determine tangent slopes for polynomials, including monomials up to the fourth power. This allowed him to derive rules for the rate of change of expressions like those involving x to the power of 1 through 4, applied in problems such as optical paths and curve tangents, as outlined in his correspondence and treatises like the Methodus ad disquirendam maximam et minimam. His work emphasized geometric adequacy over rigorous limits, providing practical tools for power-based optimization.^[17]^[18] By the 1670s, Isaac Barrow refined these ideas in his Lectiones geometricae, offering geometric proofs for the power rule specifically for integer exponents through his method of tangents. Barrow constructed differential triangles and used properties of subtangents—segments along the axis related to the tangent's direction—to demonstrate the slope for curves of the form y = x^n, generalizing the relation where the tangent's projection scales with n times the base segment. His proofs for powers like the square and cube involved aligning ordinates and parallelograms to compare infinitesimal changes, bridging pure geometry with emerging algebraic notation by expressing ratios in symbolic terms alongside diagrams. This geometric rigor for integer powers highlighted the inverse link between tangents and areas, paving the way for formalization.^[19]^[15] These 17th-century contributions by Cavalieri, Fermat, and Barrow represented a shift from purely geometric intuition to proto-algebraic methods for powers, setting the stage for Isaac Newton and Gottfried Wilhelm Leibniz to synthesize them into the broader framework of calculus.^[15]

Modern Formulation

The modern formulation of the power rule emerged in the late 17th century as part of the independent inventions of calculus by Isaac Newton and Gottfried Wilhelm Leibniz, who stated the rule explicitly for integer and rational exponents using their respective frameworks of fluxions and differentials.^[20]^[21] Newton developed the power rule around 1669 in his manuscript De Analysi per Aequationes Numero Terminorum Infinitas, deriving the fluxion of y = a x^{m/n} as \dot{y} = a \frac{m}{n} x^{m/n - 1}, which aligns with the modern derivative form \frac{d}{dx} x^r = r x^{r-1} for rational r. Leibniz formalized a similar rule in his 1684 publication Nova Methodus pro Maximis et Minimis, itemque Tangentibus in Acta Eruditorum, where he computed the differential of x^k as dx^k = k x^{k-1} dx and extended it to roots via fractional exponents, such as dz = \frac{1}{k} z^{1-k} dx for z = x^{1/k}. Newton's Philosophiæ Naturalis Principia Mathematica (1687) employed the power rule implicitly in geometric arguments for orbital mechanics, though presented without explicit fluxional notation to avoid controversy.^[20]^[21] In the 18th century, Leonhard Euler integrated the power rule into the study of infinite series, treating functions as power series expansions where term-by-term differentiation applies the rule repeatedly.^[22] In Introductio in Analysin Infinitorum (1748), Euler represented analytic functions as f(x) = \sum_{n=0}^{\infty} a_n x^n, and the derivative f'(x) = \sum_{n=1}^{\infty} n a_n x^{n-1} follows directly from the power rule, enabling rigorous manipulation of transcendental functions like exponentials and logarithms through series. The 19th century brought refinements through rigorous limit-based definitions, with Augustin-Louis Cauchy and Karl Weierstrass establishing the foundations needed for extending the power rule to real exponents.^[23]^[24] Cauchy's Cours d'Analyse de l'École Polytechnique (1821) defined limits and derivatives precisely, proving the power rule for rational exponents via the binomial theorem and limits, while using power series to analyze convergence and continuity. Weierstrass's lectures in the 1850s–1860s introduced epsilon-delta proofs for limits, continuity, and differentiability, allowing the power rule for arbitrary real exponents r through the definition x^r = e^{r \ln x} and chain rule application, with the derivative r x^{r-1} verified rigorously. By the mid-19th century, the power rule appeared routinely in calculus textbooks, reflecting its standardization in analytical education. Charles Davies's Elements of the Differential and Integral Calculus (1837) taught the rule using limits for polynomial differentiation, emphasizing practical computation.^[25] Full proofs for real exponents, grounded in the exponential-logarithmic definition, became common by the late 1800s in advanced texts, solidifying the rule's place in rigorous analysis.^[24]

Applications

In Calculus

In calculus, the power rule extends naturally to higher-order derivatives, enabling the computation of successive derivatives of power functions. For a function f(x) = x^r where r is a real number, the first derivative is f'(x) = r x^{r-1}. Applying the power rule repeatedly yields the nth derivative as f^{(n)}(x) = r(r-1)\cdots(r-n+1) x^{r-n}, where the product r(r-1)\cdots(r-n+1) is the falling factorial.^[26] If r is a non-negative integer and n > r, the nth derivative is zero, reflecting that polynomials of degree r have vanishing derivatives beyond order r. This iterative application underscores the power rule's role in analyzing the smoothness and behavior of functions through their derivative hierarchies.^[27] The power rule is instrumental in the construction and manipulation of Taylor series, which approximate functions via their derivatives at a point. In a Taylor series expansion of f(x) around a, the coefficients are given by f^{(k)}(a)/k!, and for power functions, these coefficients arise directly from repeated applications of the power rule, producing terms like r(r-1)\cdots(r-k+1) a^{r-k}/k!.^[28] Moreover, the power rule justifies term-by-term differentiation of power series, preserving convergence within the radius of convergence and yielding the series for the derivative function; for instance, differentiating \sum c_k (x-a)^k term by term applies the power rule to each monomial, resulting in \sum k c_k (x-a)^{k-1}. This property facilitates deriving series for related functions, such as integrals or compositions, enhancing analytical techniques in calculus.^[29] For polynomials, the power rule combines with the sum and constant multiple rules to differentiate any polynomial efficiently, while ties to the product and quotient rules handle more complex expressions. Consider differentiating p(x) = (x^2 + 3x)^4; expanding via the binomial theorem gives p(x) = \sum_{k=0}^4 \binom{4}{k} (x^2)^k (3x)^{4-k} = x^8 + 12 x^7 + 54 x^6 + 108 x^5 + 81 x^4, and applying the power rule term by term yields p'(x) = 8x^7 + 84 x^6 + 324 x^5 + 540 x^4 + 324 x^3.^[30] This approach leverages the power rule's simplicity on monomials, avoiding direct use of the chain rule for the composite form and illustrating its foundational utility in polynomial calculus. For quotients of polynomials, the power rule is essential in applying the quotient rule, as it is used to differentiate the numerator and denominator polynomials.^[27]

In Physics and Engineering

In kinematics, the power rule facilitates the computation of velocity and acceleration from position functions expressed as powers of time, which model various physical motions such as those under constant or variable acceleration. Consider a position function x(t) = a t^n, where a is a constant and n is the exponent; the velocity is obtained by differentiation as v(t) = \frac{dx}{dt} = n a t^{n-1}, and the acceleration follows as a(t) = \frac{dv}{dt} = n(n-1) a t^{n-2}.^[31] This approach is particularly useful for analyzing non-uniform motions, like the trajectory of objects in fields where acceleration varies with time, providing instantaneous rates essential for predicting dynamic behavior.^[31] In the context of work and energy, the power rule is employed to derive conservative forces from potential energy functions that follow power laws, linking stored energy to mechanical interactions. For the gravitational potential energy U(r) = -\frac{GMm}{r} = -k r^{-1}, where k = GMm, the radial force is F(r) = -\frac{dU}{dr} = -k (-1) r^{-2} = -\frac{k}{r^2}, directly applying the power rule to the exponent -1.^[32] This derivation yields the inverse-square law of gravitation, central to calculating work done by gravitational fields and changes in kinetic energy during motion.^[32] Engineering applications leverage the power rule in scaling laws, where physical quantities vary as powers of a characteristic length, such as radius r, to predict system performance across sizes. In fluid dynamics and process engineering, power consumption P often scales as P \propto r^3 for certain configurations like mixing or pumping operations under dimensional similarity.^[33] Differentiating this relation gives the sensitivity \frac{dP}{dr} \propto 3 r^2, quantifying how power changes with scale and informing optimizations in design, such as minimizing energy use in scaled-up reactors or pipelines.^[33] A prominent example in physics arises in orbital mechanics, where the power rule on the gravitational potential elucidates Kepler's laws. Starting from U(r) \propto r^{-1}, differentiation produces F \propto r^{-2}; balancing this with centripetal force for circular orbits yields orbital speed v = \sqrt{\frac{GM}{r}} \propto r^{-1/2}.^[34] This relation underpins Kepler's third law, T^2 \propto r^3, as Newton demonstrated using these derivatives to unify planetary motion with universal gravitation.^[34]

Generalizations

Complex Exponents

In complex analysis, the power rule extends to functions of the form f(z) = z^r, where z is a nonzero complex number and r is a complex exponent. This function is defined as f(z) = e^{r \Log z}, with \Log z denoting the principal branch of the complex logarithm, given by \Log z = \ln |z| + i \Arg z where \Arg z \in (-\pi, \pi].^[35] This definition restricts the domain to the complex plane minus the non-positive real axis to ensure single-valuedness on the principal branch.^[36] The derivative of f(z) follows the familiar power rule form: f'(z) = r z^{r-1}, provided f is analytic at z. This holds because differentiation via the chain rule yields f'(z) = e^{r \Log z} \cdot r \cdot \frac{1}{z} = r z^{r-1}, valid on any branch where the function is holomorphic.^[35] Analytic continuation allows extension beyond the principal branch, but the expression remains multi-valued due to the periodicity of the exponential, requiring careful selection of branches for consistency.^[35] A key challenge arises from branch points at z = 0 and z = \infty, where the function fails to be single-valued or analytic. At z = 0, encircling the origin changes the argument by $2\pi i, altering \Log z and thus z^r. Similarly, at infinity, substituting w = 1/z reveals a branch point at w = 0, confirming the singularity.^[37] For example, when r = 1/2, z^{1/2} represents the square root function with two branches, typically separated by a branch cut along the negative real axis; traversing around z = 0 switches between the positive and negative roots.^[37]

Multivariable and Vector Extensions

The power rule generalizes to functions of multiple variables through partial differentiation, where each variable is treated as independent while holding the others constant. For a function of the form f(\mathbf{x}) = x_1^r x_2^s \cdots x_m^t, where \mathbf{x} = (x_1, \dots, x_m) \in \mathbb{R}^m, the partial derivative with respect to x_i follows the standard power rule applied to that component:

\frac{\partial f}{\partial x_i} = r_i x_i^{r_i - 1} \prod_{j \neq i} x_j^{r_j},

with r_i denoting the exponent on x_i. This extension preserves the form of the single-variable rule and applies to monomials or products thereof, enabling differentiation of homogeneous functions via Euler's theorem in multivariable settings.^[38] In vector calculus, the power rule applies component-wise to radial scalar functions, yielding explicit expressions for the gradient. Consider f(\mathbf{x}) = \|\mathbf{x}\|^r for \mathbf{x} \in \mathbb{R}^n \setminus \{\mathbf{0}\} and r \neq 0, where \|\mathbf{x}\| = (\sum_{i=1}^n x_i^2)^{1/2}. The gradient is

\nabla f(\mathbf{x}) = r \|\mathbf{x}\|^{r-2} \mathbf{x},

derived by recognizing f as a composition involving the norm and applying the chain rule alongside the power rule to each coordinate. This formula is foundational for analyzing radial potentials in physics, such as gravitational or electrostatic fields where r = -1. For radial vector fields \mathbf{F}(\mathbf{r}) = g(r) \mathbf{r} with r = \|\mathbf{r}\|, the divergence in \mathbb{R}^3 leverages spherical coordinates, where the power rule simplifies computation: \nabla \cdot \mathbf{F} = \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 g(r)) = r \frac{d g}{dr} + 3 g(r). For power-law forms g(r) = c r^{k-1}, this yields \nabla \cdot \mathbf{F} = (k + 2) c r^{k-1}, illustrating the rule's utility in coordinate-adapted derivatives.^[39] Fractional powers appear in solutions to partial differential equations (PDEs), particularly in self-similar forms. For the heat equation \partial_t u = \Delta u in \mathbb{R}^n, the fundamental solution is u(\mathbf{x}, t) = (4 \pi t)^{-n/2} \exp(-\|\mathbf{x}\|^2 / (4 t)), where the exponent n/2 is fractional for odd n (e.g., r = 1/2 in one dimension), and differentiation via the power rule confirms its properties under the Laplacian. This structure arises from scaling arguments and Gaussian integrals, with the power rule facilitating verification of diffusion behavior. In the complex multivariable setting, for holomorphic functions analytic in several variables, the power rule extends analogously to partial derivatives of monomials, subject to the Cauchy-Riemann equations in higher dimensions.

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[PDF] 18.04 Complex analysis with applications - MIT Mathematics
A function f(z) is analytic if it has a complex derivative f0(z). In general, the rules for computing derivatives will be familiar to you from single variable.
[33]
https://www.aiche.org/resources/publications/cep/2019/october/applying-scaling-laws-process-engineering
[34]
Rules of calculus - multivariate
Use the power rule on the following function to find the two partial derivatives: The composite function chain rule notation can also be adjusted for the ...
[35]
[PDF] Gradient, Divergence, Curl and Related Formulae - UT Physics
∇P(r) = dP dr ∇r. = dP dr r. (10) where the second equality follows from the gradient of the radius r being the unit vector r in the radial direction. This ...