produktregeln

Produktregeln, or the product rule, is a fundamental rule in calculus for differentiating the product of functions. If f and g are differentiable at a point x, then the derivative of their product h(x) = f(x) g(x) is h′(x) = f′(x) g(x) + f(x) g′(x). This rule follows from the limit definition of the derivative and is often derived using the distributive property and the chain rule.

Generalization: For a product of n differentiable functions f1, f2, ..., fn, the derivative is d/dx ∏_{i=1}^n f_i(x) = ∑_{k=1}^n f_k′(x) ∏_{i≠k} f_i(x). In other words, the derivative is the sum of each function’s derivative times the product of the remaining functions. If any factor is constant, the rule reduces to the familiar two-function case.

Example: d/dx [ x^2 sin x ] = 2x sin x + x^2 cos x.

Notes: The rule is named after Leibniz. It is widely used in physics, engineering, economics, and any situation involving rates of change of products. The product rule works in conjunction with the chain rule to differentiate more complex compositions and products.