This article is about the derivative of a product. For the relation between derivatives of 3 dependent variables, see Triple product rule. For a counting principle in combinatorics, see Rule of product. For conditional probabilities, see Chain rule (probability).
Geometric illustration of a proof of the product rule
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
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
and this is indeed the differential form of the product rule. If we divide through by the differential dx, we obtain
Suppose we want to differentiate f(x) = x2 sin(x). By using the product rule, one gets the derivative f′(x) = 2x sin(x) + x2 cos(x) (since the derivative of x2 is 2x 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 f(x) is a differentiable function, then cf(x) is also differentiable, and its derivative is (cf)′(x) = cf′(x). 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.)
Proofs
Limit definition of derivative
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.
The fact that follows from the fact that differentiable functions are continuous.
Linear approximations
By definition, if are differentiable at , then we can write linear approximations:
and
where the error terms are small with respect to h: that is, also written. Then:
The "error terms" consist of items such as and which are easily seen to have magnitude Dividing by and taking the limit gives the result.
In the context of Lawvere's approach to infinitesimals, let dx be a nilsquare infinitesimal. Then du = u′ dx and dv = v ′ dx, so that
since
Taking log
Let . Taking natural log on both sides, . Differentiating:
Generalizations
Product of more than two factors
The product rule can be generalized to products of more than two factors. For example, for three factors we have
For a collection of functions , 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
Using that the logarithm of a product is the sum of the logarithms of the factors, the sum rule for derivatives gives immediately
The last above expression of the derivative of a product is obtained by multiplying both members of this equation by the product of the
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:
Applied at a specific point x, the above formula gives:
Furthermore, for the nth derivative of an arbitrary number of factors, one has a similar formula with multinomial coefficients:
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,
This is also a special case of the product rule for bilinear maps in Banach space.
Derivations in abstract algebra and differential geometry
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.
Among the applications of the product rule is a proof that
when n is a positive integer (this rule is true even if n is not positive or is not an integer, but the proof of that must rely on other methods). 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. If the rule holds for any particular exponent n, then for the next value, n + 1, we have
Therefore, if the proposition is true for n, it is true also for n + 1, and therefore for all natural n.