Which rule is specifically used for differentiating integrals with variable limits?

The rule specifically used for differentiating integrals with variable limits is Leibnitz's Rule. This rule is essential when dealing with integrals where the limits themselves are functions of a variable. It provides a systematic way to differentiate the integral in situations where both the integrand and the limits are functions of that variable.

Leibnitz's Rule states that if you have an integral from a function ( a(t) ) to ( b(t) ) of a function ( f(x, t) ), then the derivative of this integral with respect to ( t ) can be computed using the formula:

[

\frac{d}{dt} \left( \int_{a(t)}^{b(t)} f(x, t) , dx \right) = f(b(t), t) \cdot b'(t) - f(a(t), t) \cdot a'(t) + \int_{a(t)}^{b(t)} \frac{\partial f}{\partial t}(x, t) , dx

]

This expression incorporates not only the evaluation of the function at the variable limits but also includes the effect of the limits changing with ( t ), which is crucial for accurately computing the derivative.