Product rule

In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated as

or in the Leibniz notation

In differentials notation, this can be written as

In Leibniz notation, the derivative of the product of three functions (not to be confused with Euler's triple product rule) is

Discovery of this rule is credited to Gottfried Leibniz, who demonstrated it using differentials. (However, Child (2008) 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

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 ${\displaystyle f(x)g(x+\Delta {x})-f(x)g(x+\Delta {x})}$ (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.

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