Leibnizszabály

Leibnizszabály refers to the product rule for differentiation in calculus. It is named after the German mathematician Gottfried Wilhelm Leibniz. The rule states that the derivative of a product of two functions is equal to the sum of the product of the derivative of the first function and the second function, and the product of the first function and the derivative of the second function.

In mathematical notation, if we have two differentiable functions, u(x) and v(x), their product is f(x) = u(x)v(x). The Leibnizszabály, or product rule, states that the derivative of f(x) with respect to x, denoted as f'(x) or d/dx[u(x)v(x)], is given by:

f'(x) = u'(x)v(x) + u(x)v'(x)

This rule is a fundamental concept in calculus and is used extensively in differentiating more complex functions. It can be extended to the derivative of a product of more than two functions. For instance, if we have three functions, u(x), v(x), and w(x), the derivative of their product is u'(x)v(x)w(x) + u(x)v'(x)w(x) + u(x)v(x)w'(x).

The Leibnizszabály is essential for solving various calculus problems, including finding derivatives of algebraic expressions, trigonometric functions, and exponential functions when they appear as products. Its derivation can be understood through the limit definition of the derivative.