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In calculus, the **product rule** (or **Leibniz rule**^{[1]} 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's notation as

or in Leibniz's notation as

The rule may be extended or generalized to products of three or more functions, to a rule for higher-order derivatives of a product, and to other contexts.

Discovery of this rule is credited to Gottfried Leibniz, who demonstrated it using differentials.^{[2]} (However, J. M. Child, a translator of Leibniz's papers,^{[3]} argues that it is due to Isaac Barrow.) Here is Leibniz's argument: Let *u*(*x*) and *v*(*x*) be two differentiable functions of *x*. Then the differential of *uv* is

Since the term *du*·*dv* is "negligible" (compared to *du* and *dv*), Leibniz concluded that

- Suppose we want to differentiate By using the product rule, one gets the derivative (since the derivative of is and the derivative of the sine function is the cosine function).
- One special case of the product rule is the constant multiple rule, which states: if c is a number, and is a differentiable function, then is also differentiable, and its derivative is This follows from the product rule since the derivative of any constant is zero. This, combined with the sum rule for derivatives, shows that differentiation is linear.
- The rule for integration by parts is derived from the product rule, as is (a weak version of) the quotient rule. (It is a "weak" version in that it does not prove that the quotient is differentiable but only says what its derivative is
*if*it is differentiable.)

Let *h*(*x*) = *f*(*x*)*g*(*x*) and suppose that f and g are each differentiable at x. We want to prove that h is differentiable at x and that its derivative, *h′*(*x*), is given by *f′*(*x*)*g*(*x*) + *f*(*x*)*g′*(*x*). To do this, (which is zero, and thus does not change the value) is added to the numerator to permit its factoring, and then properties of limits are used.

By definition, if are differentiable at , then we can write linear approximations:

This proof uses the chain rule and the quarter square function with derivative . We have:

The product rule can be considered a special case of the chain rule for several variables, applied to the multiplication function :

Let *u* and *v* be continuous functions in *x*, and let *dx*, *du* and *dv* be infinitesimals within the framework of non-standard analysis, specifically the hyperreal numbers. Using st to denote the standard part function that associates to a finite hyperreal number the real infinitely close to it, this gives

In the context of Lawvere's approach to infinitesimals, let be a nilsquare infinitesimal. Then and , so that

Let . Taking the absolute value of each function and the natural log of both sides of the equation,

The product rule can be generalized to products of more than two factors. For example, for three factors we have

The logarithmic derivative provides a simpler expression of the last form, as well as a direct proof that does not involve any recursion. The *logarithmic derivative* of a function f, denoted here Logder(*f*), is the derivative of the logarithm of the function. It follows that

It can also be generalized to the general Leibniz rule for the *n*th derivative of a product of two factors, by symbolically expanding according to the binomial theorem:

Applied at a specific point *x*, the above formula gives:

Furthermore, for the *n*th derivative of an arbitrary number of factors, one has a similar formula with multinomial coefficients:

For partial derivatives, we have^{[4]}

Suppose *X*, *Y*, and *Z* are Banach spaces (which includes Euclidean space) and *B* : *X* × *Y* → *Z* is a continuous bilinear operator. Then *B* is differentiable, and its derivative at the point (*x*,*y*) in *X* × *Y* is the linear map *D*_{(x,y)}*B* : *X* × *Y* → *Z* given by

This result can be extended^{[5]} to more general topological vector spaces.

The product rule extends to various product operations of vector functions on :^{[6]}

- For scalar multiplication:
- For dot product:
- For cross product of vector functions on :

There are also analogues for other analogs of the derivative: if *f* and *g* are scalar fields then there is a product rule with the gradient:

Such a rule will hold for any continuous bilinear product operation. Let *B* : *X* × *Y* → *Z* be a continuous bilinear map between vector spaces, and let *f* and *g* be differentiable functions into *X* and *Y*, respectively. The only properties of multiplication used in the proof using the limit definition of derivative is that multiplication is continuous and bilinear. So for any continuous bilinear operation,

In abstract algebra, the product rule is the defining property of a derivation. In this terminology, the product rule states that the derivative operator is a derivation on functions.

In differential geometry, a tangent vector to a manifold *M* at a point *p* may be defined abstractly as an operator on real-valued functions which behaves like a directional derivative at *p*: that is, a linear functional *v* which is a derivation,

Among the applications of the product rule is a proof that

- Differentiation of integrals – Problem in mathematics
- Differentiation of trigonometric functions – Mathematical process of finding the derivative of a trigonometric function
- Differentiation rules – Rules for computing derivatives of functions
- Distribution (mathematics) – Mathematical analysis term similar to generalized function
- General Leibniz rule – Generalization of the product rule in calculus
- Integration by parts – Mathematical method in calculus
- Inverse functions and differentiation – Calculus identity
- Linearity of differentiation – Calculus property
- Power rule – Method of differentiating single term polynomials
- Quotient rule – Formula for the derivative of a ratio of functions
- Table of derivatives – Rules for computing derivatives of functions
- Vector calculus identities – Mathematical identities

**^**"Leibniz rule – Encyclopedia of Mathematics".**^**Michelle Cirillo (August 2007). "Humanizing Calculus".*The Mathematics Teacher*.**101**(1): 23–27. doi:10.5951/MT.101.1.0023.**^**Leibniz, G. W. (2005) [1920],*The Early Mathematical Manuscripts of Leibniz*(PDF), translated by J.M. Child, Dover, p. 28, footnote 58, ISBN 978-0-486-44596-0**^**Micheal Hardy (January 2006). "Combinatorics of Partial Derivatives" (PDF).*The Electronic Journal of Combinatorics*.**13**. arXiv:math/0601149. Bibcode:2006math......1149H.**^**Kreigl, Andreas; Michor, Peter (1997).*The Convenient Setting of Global Analysis*(PDF). American Mathematical Society. p. 59. ISBN 0-8218-0780-3.**^**Stewart, James (2016),*Calculus*(8 ed.), Cengage, Section 13.2.