# Category: Theorems in calculus

Product rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange'
Fubini's theorem
In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. O
Reciprocal rule
In calculus, the reciprocal rule gives the derivative of the reciprocal of a function f in terms of the derivative of f. The reciprocal rule can be used to show that the power rule holds for negative
Differentiation of integrals
In mathematics, the problem of differentiation of integrals is that of determining under what circumstances the mean value integral of a suitable function on a small neighbourhood of a point approxima
Chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g. More precisely, if is the func
Intermediate value theorem
In mathematical analysis, the intermediate value theorem states that if is a continuous function whose domain contains the interval [a, b], then it takes on any given value between and at some point w
Taylor's theorem
In calculus, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a polynomial of degree k, called the kth-order Taylor polynomial. For a smooth functio
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field
Uniqueness theorem for Poisson's equation
The uniqueness theorem for Poisson's equation states that, for a large class of boundary conditions, the equation may have many solutions, but the gradient of every solution is the same. In the case o
Generalized Stokes theorem
In vector calculus and differential geometry the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, is a statement ab
Fermat's theorem (stationary points)
In mathematics, Fermat's theorem (also known as interior extremum theorem) is a method to find local maxima and minima of differentiable functions on open sets by showing that every local extremum of
General Leibniz rule
In calculus, the general Leibniz rule, named after Gottfried Wilhelm Leibniz, generalizes the product rule (which is also known as "Leibniz's rule"). It states that if and are -times differentiable fu
Inverse function theorem
In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that i
Bioche's rules
Bioche's rules, formulated by the French mathematician (1859–1949), are rules to aid in the computation of certain indefinite integrals in which the integrand contains sines and cosines. In the follow
Divergence theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a theorem which relates the flux of a vector field through a closed surface to the divergence of
Differentiation rules
This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus.
Extreme value theorem
In calculus, the extreme value theorem states that if a real-valued function is continuous on the closed interval , then must attain a maximum and a minimum, each at least once. That is, there exist n
Implicit function theorem
In mathematics, more specifically in multivariable calculus, the implicit function theorem is a tool that allows relations to be converted to functions of several real variables. It does so by represe
Mean value theorem
In mathematics, the mean value theorem (or Lagrange theorem) states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel
Darboux's theorem (analysis)
In mathematics, Darboux's theorem is a theorem in real analysis, named after Jean Gaston Darboux. It states that every function that results from the differentiation of another function has the interm
Monotone convergence theorem
In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences (sequences that are non-decreasing or
Squeeze theorem
In calculus, the squeeze theorem (also known as the sandwich theorem, among other names) is a theorem regarding the limit of a function that is trapped between two other functions. The squeeze theorem
Pappus's centroid theorem
In mathematics, Pappus's centroid theorem (also known as the Guldinus theorem, Pappus–Guldinus theorem or Pappus's theorem) is either of two related theorems dealing with the surface areas and volumes
Quotient rule
In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let where both f and g are differentiable and The quotient rule st
L'Hôpital's rule
In calculus, L'Hôpital's rule or L'Hospital's rule (French: [lopital], English: /ˌloʊpiːˈtɑːl/, loh-pee-TAHL), also known as Bernoulli's rule, is a theorem which provides a technique to evaluate limit
Power rule
In calculus, the power rule is used to differentiate functions of the form , whenever is a real number. Since differentiation is a linear operation on the space of differentiable functions, polynomial
Triple product rule
The triple product rule, known variously as the cyclic chain rule, cyclic relation, cyclical rule or Euler's chain rule, is a formula which relates partial derivatives of three interdependent variable
Fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a functio
Linearity of differentiation
In calculus, the derivative of any linear combination of functions equals the same linear combination of the derivatives of the functions; this property is known as linearity of differentiation, the r
Integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the pro
Integration using Euler's formula
In integral calculus, Euler's formula for complex numbers may be used to evaluate integrals involving trigonometric functions. Using Euler's formula, any trigonometric function may be written in terms
Inverse function rule
In calculus, the inverse function rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function f in terms of the derivative of f. More precisely, if the in
Rolle's theorem
In calculus, Rolle's theorem or Rolle's lemma essentially states that any real-valued differentiable function that attains equal values at two distinct points must have at least one stationary point s
Cantor's intersection theorem
Cantor's intersection theorem refers to two closely related theorems in general topology and real analysis, named after Georg Cantor, about intersections of decreasing nested sequences of non-empty co
Green's theorem
In vector calculus, Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C. It is the two-dimensional special case of Stokes'
Integral of inverse functions
In mathematics, integrals of inverse functions can be computed by means of a formula that expresses the antiderivatives of the inverse of a continuous and invertible function , in terms of and an anti