Product rule

For two functions, it may be stated in Lagrange'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 "infinitesimals" (a precursor to the modern differential).

[2] (However, J. M. Child, a translator of Leibniz's papers,[3] argues that it is due to Isaac Barrow.)

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

(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.

This proof uses the chain rule and the quarter square function

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

This was essentially Leibniz's proof exploiting the transcendental law of homogeneity (in place of the standard part above).

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

Applying properties of the absolute value and logarithms,

Taking the logarithmic derivative of both sides and then solving for

Note: Taking the absolute value of the functions is necessary for the logarithmic differentiation of functions that may have negative values, as logarithms are only real-valued for positive arguments.

, which justifies taking the absolute value of the functions for logarithmic differentiation.

The logarithmic derivative provides a simpler expression of the last form, as well as a direct proof that does not involve any recursion.

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

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

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[6] to more general topological vector spaces.

:[7] 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.

This is also a special case of the product rule for bilinear maps in Banach space.

In abstract algebra, the product rule is the defining property of a derivation.

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,

Generalizing (and dualizing) the formulas of vector calculus to an n-dimensional manifold M, one may take differential forms of degrees k and l, denoted

The proof is by mathematical induction on the exponent n. If n = 0 then xn is constant and nxn − 1 = 0.

The rule holds in that case because the derivative of a constant function is 0.