Proof of the constant multiple rule
Builds on Linearity of differentiation
Proof
We deduce the rule from the linearity of differentiation.
- Apply linearity to the functions and , which are both differentiable at , with and .
- For each , . So .
- Linearity gives that is differentiable at , with .
Idea
A constant multiple is the special case of a linear combination where the second coefficient is zero. So the rule is a direct case of linearity.
Reading path
To understand this page from the ground up, read these 68 pages in order. The foundations come first. The path takes the proofs Proof of the limit of a sum of sequences, Proof of the approximation property of the supremum, Proof that there is a natural number n with 1/n below any positive epsilon, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the limit of a product of sequences, Proof that the absolute value is multiplicative, Proof of the linearity of differentiation, Proof of the sequential characterization of limits, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof of the epsilon-delta form of the limit of a function, Proof that the reals are Archimedean from the completeness axiom, Proof that convergent sequences are bounded.
- Construction of the reals from Cauchy sequences concept
- Element definition
- Integers object
- Limit point of a set definition
- Ordered field definition
- Proposition concept
- Rational numbers object
- Real numbers object
- Sequence definition
- Set bounded above definition
- Set definition
- Ring of integers statement
- Limit of a function definition
- Archimedean ordered field definition
- Implication definition
- Predicate definition
- The reals form a field statement
- Real sequence definition
- Supremum definition
- Ordered pair definition
- Derivative definition
- The rationals form a field statement
- Universal quantifier definition
- Bounded sequence definition
- Limit of a sequence definition
- Completeness axiom statement
- Cartesian product definition
- Differentiable function definition
- Subset definition
- The rationals form an ordered field statement
- Convergent sequence definition
- Binary relation definition
- Order on the reals definition
- Antisymmetric relation definition
- Reflexive relation definition
- Transitive relation definition
- Interval definition
- Preorder definition
- Proof of the epsilon-delta form of the limit of a function proof
- Partial order definition
- Epsilon-delta definition of a limit statement
- Total order definition
- Proof that the reals form an ordered field, with the Cauchy sequence construction proof
- The reals form an ordered field statement
- Absolute value definition
- Proof of the approximation property of the supremum proof
- Proof of the triangle inequality for real numbers proof
- Proof that the absolute value is multiplicative proof
- Proof of the epsilon-N definition of the limit of a sequence proof
- Approximation property of the supremum statement
- Triangle inequality statement
- Absolute value is multiplicative statement
- Epsilon-N definition of the limit of a sequence statement
- Proof that the reals are Archimedean from the completeness axiom proof
- Proof of the limit of a sum of sequences proof
- Proof that convergent sequences are bounded proof
- The reals are Archimedean statement
- Proof that there is a natural number n with 1/n below any positive epsilon proof
- Limit of a sum of sequences statement
- Convergent sequences are bounded statement
- There is a natural number n with 1/n below any positive epsilon statement
- Proof of the limit of a product of sequences proof
- Proof of the sequential characterization of limits proof
- Limit of a product of sequences statement
- Sequential characterization of limits statement
- Proof of the linearity of differentiation proof
- Linearity of differentiation statement
- Proof of the constant multiple rule proof
Where this leads
1083 pages build on this one.
- Singular value decomposition
- Power series are differentiable inside the radius of convergence
- A function with zero derivative is constant
- Natural logarithm
- Exterior derivative
- A differentiable function is convex iff its derivative is increasing
- Taylor's theorem with the Cauchy remainder
- Picard–Lindelöf theorem in integral form
- Euler–Mascheroni constant
- The logarithm grows slower than any power
- Non-uniqueness for a non-Lipschitz equation
- Change of variables in multiple integrals
- Arc length parametrisation
- Derivative of sine
- Lotka–Volterra equations
- Volume of a ball in R^n
- Additivity of line integrals under concatenation
- 1 + x is at most exp x
- Reduction formula
- exp z = 1 iff z is an integer multiple of 2πi
- Chernoff bound
- Integral equation form of an initial value problem
- Rolle's theorem
- Residue at a simple pole
- Maximal solution
- First derivative test
- cos pi = -1
- Periodic orbit
- Function with unequal mixed partial derivatives
- sin x / x tends to 1
- Symmetry of second derivatives
- Arguments add under multiplication
- Uniqueness of solutions for Lipschitz vector fields
- A function with nonnegative derivative is increasing
- Angle addition formulas
- Flow of a vector field
- Holomorphic functions on a disc have primitives
- Mean value inequality
- Casorati–Weierstrass theorem
- Inverse trigonometric functions
- Power series of sine
- Divergence
- The complex exponential is 2πi-periodic
- Reduction of order
- Cauchy mean value theorem
- The real part of a holomorphic function is harmonic
- Residue theorem
- Triangularisation
- Leibniz formula for pi
- Integral of exp(-x) over the half line
- Taylor's theorem with the Peano remainder
- Variance of the normal distribution
- Pythagorean trigonometric identity
- Mean value theorem in several variables
- Taylor's theorem
- Open mapping theorem
- C^1 vector fields satisfy the Picard–Lindelöf hypotheses
- Uniform limits of holomorphic functions are holomorphic
- Maximum modulus principle
- Lagrange multiplier with one constraint
- Exponential of a sum of commuting matrices
- log x is at most x - 1
- Spectral theorem for normal matrices
- Inverse hyperbolic functions
- Integral of cosine
- Parametrised surface
- The nth roots of unity
- Ptolemy's theorem
- Poincaré lemma for one-forms on convex sets
- Angle sum of a triangle
- Derivative of arctangent
- Conservative vector field
- Laplacian as a sum of second partial derivatives
- Harmonic numbers grow like log n
- Phase plane
- Jacobian of spherical coordinates
- Asymptotic stability
- Order of a zero
- LaSalle's invariance principle
- De Moivre–Laplace theorem
- Solution formula for linear first-order equations
- Integral of the exponential function
- Spectral theorem
- Regular curve
- Power series of the exponential function
- Existence of Laurent expansions on an annulus
- Simply connected domain
- Grönwall's inequality
- A function with positive derivative is strictly increasing
- Derivatives of the hyperbolic functions
- Young's inequality for products
- Derivative of the logarithm
- Holomorphic functions are analytic
- Mean of the Poisson distribution
- Picard's great theorem
- Power series of cosine
- Integration by parts
- Eigenvalue method for linear systems
- Euler's identity
- Lyapunov stability
- Power series of the complex logarithm
- Area of a region as a line integral
- Mean of the normal distribution
- Liouville's formula
- Linearisation at an equilibrium
- Determinant of a matrix exponential
- Stokes' theorem
- Differentiation under the integral sign
- Moment-generating function of the normal distribution
- Curl-free vector field on a convex set is conservative
- Arcsine
- Mean value theorem
- Solutions for complex conjugate roots
- Global existence for globally Lipschitz vector fields
- Weak law of large numbers
- Path independence of line integrals
- Tangent plane and normal vector of a surface
- Hölder's inequality for sums
- Goursat's theorem
- Polar form
- Riemann's theorem on removable singularities
- Trajectory
- Uniform limits commute with integration
- Implicit function theorem
- Zero derivative on a connected open set implies constant
- pi
- Rectifiable curve
- Hamiltonian system
- Euler product formula
- Hyperbolic equilibrium point
- Harmonic conjugate
- Classification of complex quadratic forms
- exp(2 pi i/n) is a primitive root of unity
- Impossibility of doubling the cube
- Uniform convergence of derivatives allows term-by-term differentiation
- Integrating factor for a non-exact equation
- Closed curve
- Integration by substitution
- Rouché's theorem
- The exponential function is convex
- Tangent vector of a curve
- Strong law of large numbers
- Cauchy's integral formula
- Spectral theorem for Hermitian matrices
- Basel problem
- Surface integral
- Number of nth roots of unity
- Blow-up in finite time
- Trigonometric substitution
- Limit cycle
- Liouville's theorem
- Orientable surface
- Central limit theorem
- Poincaré–Bendixson theorem
- Impossibility of trisecting the angle
- The exponential function is its own derivative
- Cauchy's integral formula for derivatives
- Closed and exact forms
- Derivative vanishes at a local extremum
- Divergence of a curl is zero
- The gamma function extends the factorial
- Scalar potential
- Heron's formula
- Grönwall bound
- Reversing a path changes the sign of a line integral
- Second derivative test
- Continuous dependence on initial data
- Cylindrical coordinates
- Inscribed angle theorem
- Derivative of arcsine
- Curl
- General solution of an inhomogeneous linear equation
- Solution space of a homogeneous linear equation has dimension n
- Curl-free field that is not conservative
- Angles in the same segment are equal
- Area under a curve
- Integral of sine
- Second derivative test in several variables
- Divergence theorem on a rectangle in the plane
- Dirichlet integral
- Fundamental system of solutions
- Taylor's theorem with the Lagrange remainder
- Solutions for distinct real roots
- Generalised Stokes theorem
- Green's theorem
- Solutions for repeated roots
- Mean value property
- Polar decomposition
- Lyapunov's stability theorem
- Logarithm of a quotient
- Taylor's theorem in several variables
- Euler method
- Vector calculus identities
- Parametrised curve
- Residue
- Underdamped
- Cauchy's estimates
- Fundamental theorem of algebra
- Estimate for approximate solutions
- L'Hôpital's rule
- Divergence theorem
- Laplace transform of a derivative
- Integral of 1/x
- Logarithm of a product
- Solving initial value problems with the Laplace transform
- Isolated zeros theorem
- Inverse function theorem
- Stirling's formula
- The exponential function is strictly increasing
- Continuous partial derivatives imply differentiability
- Derivative of x^a
- Quadratic formula
- Curl of a gradient is zero
- Line integral of a scalar field
- Schwarz lemma
- Wronskian test for linear independence of solutions
- Fundamental matrix
- Test for exactness
- One-form
- Liouville's theorem on Diophantine approximation
- Vector field
- Classification of equilibria of planar linear systems
- Principal part
- Implicit function theorem for a surjective derivative
- Fundamental theorem of calculus
- Schwarz's theorem for C^2 functions
- Characteristic function of the normal distribution
- Discrete Grönwall inequality
- Exterior angle theorem
- Second Borel–Cantelli lemma
- Sum of independent normal random variables
- Angle-angle similarity
- Classification of isometries of the plane
- Second-order Taylor expansion
- Sine is 2π-periodic
- A function with bounded derivative is Lipschitz
- Picard–Lindelöf theorem
- Law of sines
- Fermat's theorem on stationary points
- Argument principle
- The constructible numbers form a field closed under square roots
- Jordan's lemma
- Principal axis theorem
- Cumulant generating function
- Integral of a power
- Darboux's theorem
- Holomorphic functions are infinitely differentiable
- Differential form
- Irrationality of pi
- Implicit differentiation
- pi is greater than 3
- Arccosine
- Picard iteration
- Polar coordinates
- Logarithm is the inverse of the exponential
- Damped harmonic oscillator
- Wedderburn's little theorem
- Spherical coordinates
- Power-reduction formula for cosine
- Wedge product of forms
- Variation of constants formula for linear systems
- Hartman–Grobman theorem
- Derivative of tangent
- Gradient theorem
- Pendulum equation
- Irreducibility of the cyclotomic polynomials over the rationals
- Stability of linear systems via eigenvalues
- Principal value of the argument
- First integral
- Hurwitz's theorem
- The degree of a constructible number is a power of two
- Moments are derivatives of the moment-generating function at zero
- C^1 functions are strictly differentiable
- Power series of arctangent
- Phase portrait
- Integration in polar coordinates
- C^k function characterised by continuous derivatives
- Term-by-term differentiation
- Bounded derivative on a convex set implies Lipschitz
- Real exponentiation
- Derivative of cosine
- Logarithm of a power
- Morera's theorem
- Improper integral of x^p on an unbounded interval
- Abel's identity
- Lagrange multipliers
- Line integral as an integral over the parameter interval
- Harmonic function
- Tangent half-angle substitution
- Integration by partial fractions
- x^2 sin(1/x) is differentiable with discontinuous derivative
- Taylor's theorem with the integral remainder
- Sylvester's criterion
- Volume of a ball in R^3
- Hoeffding's inequality
- Surface area
- The exterior derivative squares to zero
- Forced oscillation and resonance
- Impossibility of squaring the circle
- Wallis product
- Poisson limit theorem
- Laplacian
- Method of undetermined coefficients
- The principal logarithm is holomorphic off the branch cut
- Second fundamental theorem of calculus
- Gauss–Wantzel theorem
- Variation of parameters
- Identity theorem
- Term-by-term integration
- Matrix exponential as the solution of a linear system
- Harmonic oscillator
- Order of a pole
- Principal branch
- Existence of nth roots of complex numbers
- Flux through a surface
- Fundamental theorem of contour integration
- Line integral of a vector field
- Homotopy invariance of line integrals of closed forms
- Mercator series
- Diffeomorphism
- A function with nonnegative second derivative is convex
- Local homeomorphism from the inverse function theorem
- Constant multiple rule
- p-series test
- The logarithm is strictly concave
- Complex power
- Morse–Smale system
- Proof of Taylor's theorem by repeated use of L'Hôpital's rule
- Proof of the argument principle by the residue theorem
- Geodesic curvature
- Proof of Wedderburn's little theorem by the class equation and cyclotomic polynomials
- Proof of uniqueness of solutions for Lipschitz vector fields
- Sample average approximation
- Liénard's theorem
- Spectral theorem for unbounded self-adjoint operators
- Proof of integration by parts
- Riesz projector
- Arnold tongue
- Proof of the impossibility of doubling the cube
- Emden–Chandrasekhar equation
- Simple harmonic motion
- Proof of the Lagrange multiplier rule with one constraint
- Lorentz reciprocal theorem for Stokes flow
- Proof that a function with nonnegative second derivative is convex
- Mahler measure
- Routh–Hurwitz theorem
- Proof of the integral of cosine
- Proof of the logarithm of a power
- Conformal invariance of harmonic functions
- Proof of the principal axis theorem from the spectral theorem
- Proof of the derivative of cosine by term-by-term differentiation
- Saccheri–Legendre theorem
- Lebesgue's theorem on differentiability of monotone functions
- Shell theorem
- Ergodic decomposition
- Proof that log x is at most x - 1
- Proof that a curl-free vector field on a convex set is conservative
- Hellinger–Reissner principle
- Proof of the impossibility of squaring the circle
- Proof of the Leibniz formula for pi by integrating a finite geometric sum
- Holomorphic inverse function theorem
- Minimal curvature property of natural cubic splines
- Proof of the derivatives of the hyperbolic functions
- Newton's law of universal gravitation
- Osgood uniqueness criterion
- Operational calculus
- Koksma–Hlawka inequality
- Proof of the area of a region as a line integral
- Proof of Rolle's theorem
- Proof of Abel's identity
- Phragmén–Lindelöf principle for subharmonic functions
- Riemann zeta function as a series
- Gauss's law for magnetism
- Proof of the implicit function theorem for a surjective derivative
- Carleman estimate for harmonic measure
- Proof of the test for exactness
- Stein's unbiased risk estimate
- Circle criterion
- Proof of the singular value decomposition from the spectral theorem
- Proof of the homotopy invariance of line integrals of curl-free fields by local potentials
- Cauchy–Pompeiu formula
- Proof of the implicit function theorem from the inverse function theorem
- Proof of the formula for the mean of the normal distribution
- Alfvén's frozen-in flux theorem
- Reynolds transport theorem
- Quantum Zeno effect
- Hilbert–Schmidt expansion theorem
- Proof of Morera's theorem by a local primitive
- Proof of the mean value property via a holomorphic function with real part u
- Cheng–Yau gradient estimate
- Removable singularities for harmonic functions
- Minimum principle for harmonic functions
- Method of image charges
- Lyusternik's theorem on tangent spaces
- Bridgman's thermodynamic equations
- Convergence of the Neumann series for Volterra kernels
- Displacement current
- Markov partition
- Augmented Lagrangian method
- Chandrasekhar's white dwarf equation
- Surface equivalence principle
- Durand–Kerner method
- Hardy's uncertainty principle
- Discretization of linear systems
- Canonical coordinates
- Jensen's formula
- Dedekind ellipsoid
- Proof of the solution formula for linear first-order equations
- Center manifold theorem
- Hydrostatic equilibrium
- Maxwell stress tensor
- Proof of term-by-term differentiation
- Proof of Young's inequality for products
- Kinetic energy
- Smoothing by convolution
- Tidal force
- Proof of the solutions for complex conjugate roots
- Proof of the integral of a power
- Limit set
- Céa's lemma
- Anharmonic oscillator
- Affine differential geometry
- Weierstrass product for the gamma function
- Consistency of the maximum likelihood estimator
- Monodromy theorem
- Artin–Schreier theorem
- Perron's irreducibility criterion
- Proof of Sylvester's criterion by induction and completing the square
- Proof of the Schwarz lemma by the maximum modulus principle
- Proof of the fundamental theorem of algebra by Liouville's theorem
- Proof of the Laplacian as a sum of second partial derivatives
- Proof of the Wronskian test for linear independence of solutions
- Proof that the solution space of a homogeneous linear equation has dimension n
- Asymptotic equipartition property
- Phonon dispersion in lattices
- Force-free magnetic field
- Potential energy
- Volterra's principle
- Friction
- Stein interpolation theorem
- Bilinear series of a symmetric kernel
- Kobayashi metric
- Proof of Euler's identity
- Choquet-Bruhat–Geroch theorem on maximal developments
- Clapeyron's theorem
- Magnetic braking
- Proof of the description of the nth roots of unity
- Proof of the Lagrange multiplier theorem by the implicit function theorem
- Isoperimetric problem of the calculus of variations
- Harmonic majorant
- Littlewood–Paley inequality
- Vanishing mean oscillation
- Proof of the power series of sine
- Spectral decomposition theorem
- Stein's lemma
- Finite Desarguesian planes are Pappian
- Black hole stability problem
- Gagliardo–Nirenberg interpolation inequality
- Virial theorem
- Bertrand's test
- Tests of general relativity
- Gamma function as Mellin transform of the exponential
- Milankovitch cycles
- Stellar structure
- Proof of the integral of the exponential function
- Splitting circle method
- Modified Newtonian dynamics
- Kelvin–Helmholtz mechanism
- Source–channel separation theorem
- Hohmann transfer orbit
- Proof that a bounded derivative on a convex set implies Lipschitz
- Local potential for a planar field with equal mixed partials
- Identity theorem in several variables
- Term-by-term differentiation of a Fourier series
- Mertens' theorems
- Method of lines
- Proof of the gradient theorem
- Peano kernel theorem
- Proof that the constructible numbers form the smallest subfield of the complex numbers closed under square roots
- Proof of the divergence theorem on a rectangle in the plane
- Stein's example
- Log-modulus of a holomorphic function is subharmonic
- Comparison theorem
- Central configuration
- Circulation
- Invariant set
- Proof that exp(2 pi i/n) is a primitive root of unity
- Electromagnetic radiation
- Proof that the logarithm is strictly concave
- Kingman's subadditive ergodic theorem
- Finite element method in structural mechanics
- Schwarzschild criterion for convection
- Lax–Milgram theorem
- Reduction of an initial value problem to a Volterra equation
- McDiarmid's inequality
- Cauchy-type integral
- Asymptotic normality of the maximum likelihood estimator
- Box–Muller transform
- Proof of the second-order Taylor expansion
- Virial mass
- von Zeipel theorem
- Typical set
- Ramanujan's series for 1/pi
- Catenary problem
- Proof that cos pi = -1 and sin pi = 0
- Shape operator
- Jacobi integral
- Brunt–Väisälä frequency
- Proof that the derivative vanishes at a local extremum
- Maximum modulus principle in several variables
- Proof that a function with nonnegative derivative is increasing
- Difference quotient
- Central difference approximation of the derivative
- Damping
- Edgeworth expansion
- Weyl's lemma
- Derivative of a convolution
- Gain and phase margin
- Uhlmann's theorem
- Galaxy rotation curve
- Bihari–LaSalle inequality
- Quadratic convergence of Newton's method
- Celestial mechanics
- Marshallian demand function
- Weak gravitational lensing
- Brelot space
- Kepler's equation
- Proof that reversing a path changes the sign of a line integral
- Proof that a C^1 map with invertible derivative is a local homeomorphism and an open map
- Nash–Moser theorem
- Poisson–Jensen formula
- Spectral theorem for compact self-adjoint operators
- Courant–Fischer characterization of characteristic values
- Bi-elliptic transfer
- Failure of Aizerman's conjecture
- Galilean invariance
- Proof that the logarithm is the inverse of the exponential
- Proof of Liouville's formula
- Utility maximization problem
- Frullani integral
- Minimum total potential energy principle
- Darboux function
- Gauss's test
- Proof of the spectral theorem for self-adjoint operators by induction on the dimension
- Proof of the derivative of the logarithm by the inverse function rule
- Second fundamental form
- Proof of differentiation under the integral sign
- Log-convexity of the gamma function
- Linear multistep method
- Lorentz oscillator model
- Mass–luminosity relation
- Proof of the discrete Grönwall inequality
- Fresnel equations
- Magnetic monopole
- Accretion disk
- Conservation of energy
- Principle of material frame-indifference
- Proof of Hoeffding's inequality via Hoeffding's lemma and the Chernoff bound
- Stiff equation
- Smoothness and analyticity of harmonic functions
- Sobolev inequality
- Proof of the path independence of line integrals
- Fáry–Milnor theorem
- Laplace transform methods in heat conduction
- Symplectic manifold
- Proof that C^1 vector fields satisfy the Picard–Lindelöf hypotheses
- Proof of Jordan's lemma by Jordan's inequality
- Mixed state
- ADM formalism
- Einstein–Infeld–Hoffmann equations
- Azuma's inequality
- Cyclotomic field
- Proof that uniform limits commute with integration
- Proof of Stokes' theorem by pulling back to Green's theorem
- Proof of the logarithm of a quotient
- Mean value theorem for divided differences
- Proof of the residue formula at a simple pole
- Implicit function theorem in Banach spaces
- Proof that holomorphic functions are infinitely differentiable
- First fundamental form
- Proof of the integral equation form of an initial value problem
- Melnikov method
- Geodetic triangulation
- Sub-mean value property
- Erdős–Kac theorem
- Proof of the residue theorem by subtracting principal parts
- Lens equation
- Strong cosmic censorship conjecture
- Proof that there are exactly n nth roots of unity
- Proof of the open mapping theorem by the maximum modulus principle
- Dipole model of the Earth's magnetic field
- Proof of Taylor's theorem with the Cauchy remainder by the mean value theorem
- Inelastic collision
- Proof that power series are differentiable inside the radius of convergence
- Euler's elastica
- Lyapunov–Schmidt reduction
- Uniqueness theorem for Poisson's equation
- Kakutani's solution of the Dirichlet problem
- Purification of a quantum state
- Gleason's theorem
- Weak cosmic censorship conjecture
- Proof of Cauchy's integral formula by a primitive of the difference quotient
- Numerical continuation
- Proof of the eigenvalue criteria for stability of linear systems via the Jordan normal form
- Converse of the mean value property
- Smoothness classes
- Real solutions from a complex eigenpair of a real matrix
- Two-body problem
- Proof of the de Moivre–Laplace theorem from the central limit theorem
- Total internal reflection
- Local existence of harmonic conjugates
- Ledoux criterion
- Proof that uniform convergence of derivatives allows term-by-term differentiation
- Non-vanishing of the zeta function on the line Re s = 1
- Inverted pendulum
- Euler–Maruyama method
- Proof that the complex exponential is 2πi-periodic
- Liénard equation
- Proof of Riemann's theorem on removable singularities
- Orbital mechanics
- Bonnor–Ebert mass
- Proof of the parametric formula for line integrals
- Tolman–Oppenheimer–Volkoff limit
- Monte Carlo integration
- Ehrenfest equations
- Post-Newtonian expansion
- Differentiation operator as inverse of integration
- Earnshaw's theorem
- Conformal radius
- Proof of Cauchy's estimates
- Delta method
- Proof of the Chernoff bound
- Regular surface
- Proof of the derivative of arctangent
- Stone's theorem on one-parameter unitary groups
- Proof that the degree of a constructible number is a power of two
- Proof of the formula for the variance of the normal distribution
- Proof of the power series of cosine
- Betti's reciprocal theorem
- Closing lemma
- Adams–Williamson equation
- Logarithmically convex function
- Sequential quadratic programming
- Richards equation
- Hartogs's theorem on separate holomorphicity
- Charge conservation
- Proof of the classification of complex quadratic forms by rescaling a diagonal form
- Harmonic measure
- Proof that a differentiable function is convex iff its derivative is increasing
- Total curvature
- Anosov flow
- Tisserand's parameter
- RLC circuit
- Superharmonic potential
- Proof that the principal logarithm is holomorphic off the branch cut
- Polytrope
- Cauchy formula for repeated integration
- Lidskii's trace formula for integral operators
- Nyquist stability criterion
- Interface conditions for electromagnetic fields
- Heteroclinic orbit
- Fenchel's theorem
- Proof that the logarithm grows slower than any power
- Proof of the angle addition formulas by differentiation
- Proof of the Euler product formula
- Hamiltonian operator
- Angular momentum
- Shannon–Hartley theorem
- Gravitational microlensing
- Borel–Carathéodory theorem
- Second quantization
- Dido's problem
- Continuity equation
- Differential-algebraic system of equations
- Cauchy integral formula on a polydisc
- Proof that exp z = 1 iff z is an integer multiple of 2πi
- Proof that moments are derivatives of the moment-generating function at zero
- Tsiolkovsky rocket equation
- Hayashi track
- Proof of Lyapunov's stability theorem
- Hermitian manifold
- Proof that holomorphic functions on a disc have primitives
- Proof of the general solution of an inhomogeneous linear equation
- Stellar pulsation
- Euler's three-body problem
- Phragmén–Lindelöf principle
- Smale's problems on dynamics
- Proof of the estimate for approximate solutions by a first-crossing argument
- Matsubara frequency
- Proof of the sum of independent normal random variables by characteristic functions
- Proof of the second derivative test
- Proof of the angle sum of a triangle with sines and cosines
- Conservation of angular momentum
- Kramers–Kronig relations
- Thermodynamic square
- Proof of the irreducibility of the cyclotomic polynomials by reduction modulo primes
- Comparison sort lower bound
- Picard's theorem for first order equations with continuous partial derivative
- Parker solar wind model
- Proof of the Cauchy mean value theorem from Rolle's theorem
- Reduction of a first-kind Volterra equation to the second kind
- Proof that the exponential function is strictly increasing
- Spectrum of the Laplace–Beltrami operator
- Proof of Taylor's theorem with the integral remainder by the fundamental theorem of calculus
- Proof of the power-reduction formulas
- Impulse
- Floquet theory
- Backus–Gilbert method
- Proof of the central limit theorem by characteristic functions
- Sackur–Tetrode equation
- Principal stresses
- Proof of Hölder's inequality for sums
- Berry–Esseen theorem
- Interpolation error formula
- Envelope
- Proof of Darboux's theorem from the extreme value theorem
- Proof that a function with bounded derivative is Lipschitz
- Proof that C^1 functions are strictly differentiable
- Proof of Cauchy's integral formula for derivatives
- Parametric oscillator
- Momentum
- Hardy–Littlewood–Sobolev inequality
- Ampère's circuital law
- Proof of the derivative of arcsine by the inverse function rule
- Proof of the derivative of tangent by the quotient rule
- Furuta pendulum
- Topological quantum field theory
- Escape velocity
- Smooth dynamical system
- Proof that a zero derivative on a connected open set implies constant
- Double layer potential
- Proof of the existence of nth roots of complex numbers
- Muskingum routing method
- Involute
- Proof of Rouché's theorem by the argument principle and continuity
- Birkhoff's ergodic theorem
- Chandrasekhar limit
- Eleven-dimensional supergravity
- Proof of the law of sines from the law of cosines
- Proof of Schwarz's theorem for C^2 functions by the mean value theorem
- Wick rotation
- Tidal acceleration
- Gravitational deflection of light
- Runge–Kutta methods
- Eckart–Young theorem
- Proof of the power series of the complex logarithm
- Poincaré map
- Hölder's inequality
- Poincaré inequality
- Maclaurin spheroid
- Coefficient of restitution
- Order of a zero or pole
- Strang splitting
- Q factor
- Proof of the volume of a ball in R^n by slicing
- Conformal modulus
- Korn's inequality
- Proof of the variation of constants formula for linear systems
- Jacobi integral
- Differentiable dependence on parameters
- Catenary
- Lie–Trotter product formula
- Routh–Hurwitz stability criterion
- Koebe's theorem on the mean value property
- Regular value theorem
- Structural stability
- Area-minimizing current
- Random vibration
- Borel's lemma
- Viète's formula
- Kepler's laws of planetary motion
- Inverse function theorem in Banach spaces
- Curvature of a curve
- Proof of Fermat's theorem on stationary points
- Ideal chain
- Proof of the inverse function theorem by the contraction mapping principle
- Incompressible flow
- Kharitonov's theorem
- Proof of the polar decomposition from the singular value decomposition
- Poincaré–Bendixson theorem for a bounded region without equilibria
- Homoclinic bifurcation
- Maxwell relations
- Duality of Lp spaces
- Random vibration
- Born rule
- Dynamical friction
- Borel functional calculus
- Galerkin method
- Wiener–Hopf method
- Mixing length theory
- Kepler's equation
- Compactness of families of harmonic functions
- Proof that 1 + x is at most exp x
- Proof of the formula for the characteristic function of the normal distribution
- Bernstein inequalities
- Torsion of a curve
- Minkowski inequality
- Proof that a function with positive derivative is strictly increasing
- Proof of integration by substitution
- Gradient estimates for harmonic functions
- Proof of the second Borel–Cantelli lemma
- Kepler orbit
- Frank-Kamenetskii theory of thermal explosion
- Proof that sin x / x tends to 1 as a derivative at 0
- Osgood's lemma
- Earth's magnetic field
- Oseledets theorem
- Neumann formula for mutual inductance
- Cantor function
- Veblen–Young theorem
- Electric displacement field
- Proof of Ptolemy's theorem by complex numbers
- Configuration space
- Carathéodory metric
- Schmidt decomposition
- Brody hyperbolicity
- Hyperbolic set
- Parametric oscillator
- Symplectic integrator
- Proof of the second fundamental theorem of calculus
- Schwarz reflection principle
- Harnack's principle
- Operator monotone function
- Proof of global existence for globally Lipschitz vector fields by steps of uniform length
- Hill sphere
- Proof of the spectral theorem for normal matrices by deflation
- Proof of the formula for the Laplace transform of a derivative by integration by parts
- Proof of Taylor's theorem with the Lagrange remainder by the Cauchy mean value theorem
- Proof of the Poisson limit theorem
- Poisson integral formula
- Proof of Liouville's theorem on Diophantine approximation
- Nörlund–Rice integral
- Gauss's spherical harmonic analysis of the geomagnetic field
- Mercer's theorem
- Spherical astronomy
- Proof of Heron's formula from the law of cosines
- Littlewood subordination theorem
- Proof of the spectral theorem for Hermitian matrices
- Lagrange multipliers for constraints
- Roche limit
- Proof of triangularisation by induction on the size
- Noisy-channel coding theorem
- Proof that angles in the same segment are equal
- Bifurcation theory
- Proof of term-by-term integration
- Castigliano's method
- Error bound for the trapezoidal rule
- Duhamel's principle
- Gelfand–Bratu problem
- Proof of Taylor's theorem in several variables by restriction to a line
- Proof of the power series of arctangent by integrating a finite geometric sum
- Lagrange multipliers on Banach spaces
- Anharmonic oscillator
- Proof of the volume of a ball in R^3 from the n-dimensional formula
- Somigliana identity
- Generalized Lotka–Volterra equation
- Magnetic helicity
- Proof of the impossibility of trisecting the angle
- Proof that harmonic numbers grow like log n
- Annihilator method
- Mechanical similarity
- Green's identities
- Finite difference coefficients
- Least harmonic majorant
- Proof that sine and cosine are 2π-periodic with smallest period 2π
- Bohr–Mollerup theorem
- Schönberg–Chandrasekhar limit
- Magnetotellurics
- First main theorem of Nevanlinna theory
- Proof of the isolated zeros theorem by power series
- Liouville's theorem for harmonic functions
- Quantum computing
- Proof of the derivative of sine by term-by-term differentiation
- Wave function collapse
- Proof of the additivity of line integrals under concatenation
- Flory–Huggins solution theory
- Proof of Taylor's theorem with the Peano remainder
- Proof that the exponential of a sum of commuting matrices is the product of the exponentials
- Geopotential
- Elastic pendulum
- Proof of the solutions for repeated roots
- Principal component analysis
- Proof of the derivative of x^a by the chain rule
- Liouville's theorems on elliptic functions
- Proof of the fundamental theorem of contour integration
- Proof of Goursat's theorem by repeated subdivision
- N-body simulation
- Proof of Grönwall's inequality
- Proof that uniform limits of holomorphic functions are holomorphic, by Morera's theorem
- Proof of the solutions for distinct real roots
- Proof of angle-angle similarity with the law of sines
- Volterra's function
- Celestial mechanics
- Proof of L'Hôpital's rule from the Cauchy mean value theorem
- Vogt–Russell theorem
- Hu–Washizu principle
- Mild solution
- Eddington luminosity
- Whitney extension theorem
- Heun's method
- Proof of the Poincaré lemma for one-forms on convex sets
- Proof of the existence and uniqueness of the principal value of the argument
- Proof of the mean of the Poisson distribution
- Initial value formulation of general relativity
- Proof of continuous dependence on initial data by Grönwall's inequality
- Harmonic functions on Riemann surfaces
- Gauss's law
- Hill sphere
- Proof of the polar form of a complex number
- Phase space
- Proof of the identity theorem by connectedness
- Proof of the integral of sine
- An infinite product converges iff its series of logarithms converges
- Elastic collision
- Proof of the maximum modulus principle by Cauchy's integral formula
- Quintic equation
- Axiom A
- Proof that arguments add under multiplication
- Inertia tensor and principal axes
- Barnes–Hut simulation
- Bidiagonalization
- Vis-viva equation
- Covariant formulation of classical electromagnetism
- Proof of the existence and uniqueness of Laurent expansions on an annulus
- Ladyzhenskaya–Babuška–Brezzi condition
- Proof of Stirling's formula by a monotone sequence and the Wallis product
- Duffing equation
- Classical mechanics
- Proof of the Casorati–Weierstrass theorem by Riemann's theorem
- Kapitza's pendulum
- Proof of the Pythagorean trigonometric identity by differentiation
- Work–energy theorem
- Stellar dynamics
- Milloux problem
- Kelly criterion
- Proof that the exponential function is its own derivative, by term-by-term differentiation
- Liouville's theorem for planar harmonic functions
- Liénard–Chipart criterion
- Conservation of momentum
- Gauss's law for potentials
- Proof of the second derivative test in several variables
- Proof that holomorphic functions are analytic
- Proof that the divergence of a curl is zero
- Proof of the power series of the exponential function
- Variable-mass system
- Maximum principle for harmonic functions
- Parker instability
- Descartes' rule of signs
- Trigonometric integrals
- Surface plasmon polariton
- Cauchy momentum equation
- Tautological one-form
- Indirect utility function
- Magnus expansion
- Extremal length
- Geomagnetic secular variation
- Bertrand's theorem
- Central force
- Einstein radius
- Schur product theorem
- Hadamard three-circle theorem for subharmonic functions
- Fundamental theorem of space curves
- Lindelöf's maximum principle
- Helmholtz decomposition of electromagnetic fields
- Proof that the matrix exponential solves a linear system
- Basel problem via Fourier series
- Hadamard three-circle theorem
- Schwarz–Pick theorem
- Singular values of a compact operator
- Sitnikov problem
- Proof of the Gauss–Wantzel theorem
- Abel's summation formula
- Exponential stability
- Proof of the Mercator series by term-by-term differentiation and Abel's theorem
- Spherical pendulum
- Proof of the symmetry of second derivatives by a second difference
- Jury stability criterion
- Laplace transform of an integral
- Graph embedding
- Observable
- Proof of the formula for the moment-generating function of the normal distribution
- Pesin theory
- Reptation
- Slice theorem
- Proof that a function with zero derivative on an open interval is constant
- Duhamel's principle for heat conduction
- Schmidt's singular value decomposition of a kernel
- Proof that the determinant of a matrix exponential is the exponential of the trace
- Proof that the gamma function extends the factorial
- Impossibility of the classical construction problems
- Free-fall time
- Proof of the first derivative test
- Jacobi ellipsoid
- Lagrange points
- Harmonic measure as exit distribution
- Van der Pol oscillator
- Two-body relaxation time
- Stable manifold theorem
- Proof that the exterior derivative squares to zero
- Unique solvability of Volterra equations of the second kind
- Midpoint method
- Hadamard three-lines theorem
- Parker spiral
- Picard's theorem for systems with continuous partial derivatives
- Proof of the p-series test by Cauchy condensation
- Parabolic and hyperbolic Riemann surfaces
- Poisson integral for the half-space
- Retarded potential
- Polar decomposition of the deformation gradient
- Proof of the weak law of large numbers by characteristic functions
- Anosov diffeomorphism
- Analytic capacity
- Brewster's angle
- Differentiable curve
- Proof that C^k functions are characterised by continuous derivatives
- Spectral theorem for bounded normal operators
- Poynting's theorem
- Newtonian limit
- Adjoint state method
- Proof that continuous partial derivatives imply differentiability
- Strong gravitational lensing
- Lindelöf's theorem
- Harmonic space
- Proof of the vector calculus identities
- Hadamard's two-constants theorem
- Stability conjecture
- Prime end
- Monte Carlo methods in finance
- State-transition matrix
- Proof of the fundamental theorem of calculus
- Hidden-variable theory
- Channel capacity of a quantum channel
- Proof that the real part of a holomorphic function is harmonic
- Bôcher's theorem
- Einstein ring
- Madelung equations
- Circular restricted three-body problem
- Gain scheduling
- Jefimenko's equations
- Faraday's law of induction
- Proof of the mean value theorem in several variables
- Proof of the Picard–Lindelöf theorem
- Proof of the mean value theorem from Rolle's theorem
- Evolute
- Barnes G-function
- Groundwater flow equation
- Hill differential equation
- Proof of the logarithm of a product
- Automorphism group of a bounded domain is a Lie group
- Proof of the Wallis product by integrals of powers of sine
- International Geomagnetic Reference Field
- Circle average
- Rellich–Kondrachov theorem
- Proof of the integral of 1/x
- Proof that the curl of a gradient is zero
- Proof that pi is greater than 3 by the alternating series estimate
- Proof of the classification of isometries of the plane
- Worm-like chain
- Niven's proof of the irrationality of pi
- Magnetic vector potential
- Morrey's inequality
- Osculating circle
- Chudnovsky algorithm
- Bullen parameter
- Proof of Liouville's theorem by Cauchy's estimates
- Four-vertex theorem
- Euler integral for the gamma function
- Diversity index
- Normal form
- Roche lobe
- Hartogs lemma for subharmonic functions
- Proof of the inscribed angle theorem by complex numbers
- Proof of the mean value inequality
- Frenet–Serret formulas
- Runge's phenomenon
- Proof that the exponential function is strictly convex
- Proof of Hurwitz's theorem by Rouché's theorem
- Whitney–Graustein theorem
- Inductance
- Proof of the exterior angle theorem from the angle sum
- Montel's theorem
- Euler–Lagrange equation
- Residue calculus for Laplace inversion
Further reading
- Web https://en.wikipedia.org/wiki/Linearity_of_differentiation, Constant coefficient
About this page
| Tag | ZW8Z49W7NT |
|---|---|
| URI | m2://ZW8Z49W7NT |
| Kind | proof |
| Writing | W0 stub |
| MSC | 26A24 |
| Proves | Constant multiple rule |
| Uses | Linearity of differentiation |
| Changed | 4 November 2026 |
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