integraalderivative

The integraalderivative is a concept that describes the relationship between integration and differentiation, usually referring to the derivative of an integral with a variable upper limit. In standard terminology this is the derivative of the function F(x) = ∫_a^x f(t) dt with respect to x.

Definition: For a function f continuous on [a, b], the function F(x) = ∫_a^x f(t) dt is differentiable on (a, b) and F'(x) = f(x). Therefore the integraalderivative of f, in this setting, yields f(x). This illustrates the fundamental theorem of calculus, which links antiderivatives and derivatives.

Generalizations: If the integral has variable limits both as upper and lower, G(x) = ∫_{u(x)}^{v(x)} f(t) dt, then G'(x) = f(v(x)) v'(x) - f(u(x)) u'(x). The integraalderivative concept extends to parameter-dependent integrals and higher dimensions with appropriate versions of Leibniz's rule.

Examples: If f(t) = t^2 and a = 0, F(x) = ∫_0^x t^2 dt = x^3/3, and F'(x) = x^2 = f(x). Another example: G(x) = ∫_{0}^{x^2} t dt, then G'(x) = f(v(x)) v'(x) - f(u(x)) u'(x) = f(x^2)·2x = x^2·2x = 2x^3.

Note about terminology: The expression "integraalderivative" is not standard in most texts; more common phrasing is "derivative of an integral" or "Fundamental Theorem of Calculus." It may appear in non-English contexts or educational materials as a descriptive label.

Applications: In physics, engineering, and signal processing, the identity F'(x) = f(x) is used to relate accumulated quantities to instantaneous rates.