mathmadx said:Dear all, a question which has puzzled me for some days:
(Assume that all are differentiable enough times):
Calculate:
[tex]
\frac{\mathrm{d} }{\mathrm{d} x}\int_{g(x)}^{h(x)} f(x,t) dt
[/tex]
Differentiation of an integral is the process of finding the derivative of a function that is defined as the integral of another function. It involves using the fundamental theorem of calculus and the rules of differentiation to find the derivative of the integral.
Differentiation of an integral is important because it allows us to find the rate of change of a quantity that is defined by an integral. This is useful in many areas of mathematics, physics, and engineering where the rate of change is crucial in understanding a system.
The steps involved in differentiating an integral include using the fundamental theorem of calculus to rewrite the integral in terms of the original function, applying the rules of differentiation to the function, and then simplifying the resulting expression.
Yes, there are some special cases and exceptions when differentiating an integral. These include using the chain rule when the limits of integration are functions of the variable of differentiation, using the product rule when the integrand is a product of two functions, and using the quotient rule when the integrand is a quotient of two functions.
No, differentiation of an integral can only be applied to definite integrals, meaning those with specified limits of integration. Indefinite integrals, which have no specified limits, cannot be differentiated as they result in a family of functions rather than a single function.