Produktreglan

Produktreglan, or the product rule, is a fundamental theorem in differential calculus that governs the derivative of a product of differentiable functions. If f and g are differentiable with respect to x, then the derivative of their product is (f g)' = f' g + f g'.

Generalization to more factors: For a product of n differentiable functions f1(x), f2(x), ..., fn(x), the derivative is the sum over i of fi'(x) multiplied by the product of all the other functions. In particular, (∏_{i=1}^n fi(x))' = ∑_{i=1}^n fi'(x) ∏_{j≠i} fj(x).

Higher derivatives: The Leibniz rule extends to higher-order derivatives for the product of two functions: (f g)^{(n)}(x) = ∑_{k=0}^n binom(n,k) f^{(k)}(x) g^{(n-k)}(x). For products of more functions, analogous formulas exist, involving multivariate combinatorics.

Examples and applications: A simple example is differentiating f(x) = x^2 e^x, yielding f'(x) = 2x e^x + x^2 e^x. The rule is widely used in physics and engineering, for instance in cases where momentum p = m v or where a quantity is the product of time-varying factors.

History: The rule is traditionally attributed to Gottfried Wilhelm Leibniz, who introduced it in the development of differential calculus in the late 17th century. It remains a basic tool in calculus education and analysis.