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Concept
Product rule
Summary
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'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. (However, J. M. Child, a translator of Leibniz's papers, 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
which can also be written in Lagrange's notation as
Suppose we want to differentiate f(x) = x2 sin(x). By using the product rule, one gets the derivative (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 (x) = c (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 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, (x), is given by (x)g(x) + f(x)(x).
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