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A generalization of the product rule for expressing arbitrary-order derivatives of products of functions, where (n; k) is a binomial coefficient. This can also be written ...

An identity is a mathematical relationship equating one quantity to another (which may initially appear to be different).

The Leibniz harmonic triangle is the number triangle given by 1/11/2 1/21/3 1/6 1/31/4 1/(12) 1/(12) 1/41/5 1/(20) 1/(30) 1/(20) 1/5 (1) (OEIS A003506), where each fraction ...

The Leibniz integral rule gives a formula for differentiation of a definite integral whose limits are functions of the differential variable, (1) It is sometimes known as ...

Also known as the alternating series test. Given a series sum_(n=1)^infty(-1)^(n+1)a_n with a_n>0, if a_n is monotonic decreasing as n->infty and lim_(n->infty)a_n=0, then ...

The series for the inverse tangent, tan^(-1)x=x-1/3x^3+1/5x^5+.... Plugging in x=1 gives Gregory's formula 1/4pi=1-1/3+1/5-1/7+1/9-.... This series is intimately connected ...

The derivative identity d/(dx)[f(x)g(x)] = lim_(h->0)(f(x+h)g(x+h)-f(x)g(x))/h (1) = (2) = lim_(h->0)[f(x+h)(g(x+h)-g(x))/h+g(x)(f(x+h)-f(x))/h] (3) = f(x)g^'(x)+g(x)f^'(x), ...

In a set X equipped with a binary operation · called a product, the multiplicative identity is an element e such that e·x=x·e=x for all x in X. It can be, for example, the ...

An algebraic identity is a mathematical identity involving algebraic functions. Examples include the Euler four-square identity, Fibonacci identity, Lebesgue identity, and ...

There are two identities known as Catalan's identity. The first is F_n^2-F_(n+r)F_(n-r)=(-1)^(n-r)F_r^2, where F_n is a Fibonacci number. Letting r=1 gives Cassini's ...