The rate of change of a function defined as a definite integral with respect to its upper limit of integration is equal to the integrand evaluated at that upper limit. Consider a function F(x) defined by an integral whose lower limit is a constant a and whose upper limit is a variable x. Finding the derivative of F(x) effectively reverses the process of integration at the upper bound. For example, if F(x) = ax f(t) dt, then F'(x) = f(x).
This concept, a key result from the Fundamental Theorem of Calculus, provides a powerful shortcut for differentiation, particularly when dealing with functions defined in integral form. It simplifies calculations and is essential in various areas of mathematics, physics, and engineering. It facilitates solving differential equations, analyzing the behavior of solutions, and understanding the relationship between displacement, velocity, and acceleration. Furthermore, it streamlines certain proofs and computations in advanced calculus.
