The Leibnitz Rule is used for what purpose?

The Leibnitz Rule is a powerful tool used in calculus for differentiating under the integral sign. This rule allows one to compute the derivative of an integral whose limits and/or integrand depend on a variable.

Specifically, if you have an integral of the form ( f(x) = \int_{a(x)}^{b(x)} g(t, x) , dt ), the rule determines how to differentiate this expression with respect to ( x ). It states that the derivative of this integral is given by:

[

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

]

This formulation showcases how the rule captures changes in both the limits of integration and the function being integrated.

In contrast, the other purposes listed do not align with the primary function of the Leibnitz Rule. For instance, differentiating functions generally does not involve the integration process, while integrating products of functions typically pertains to integration techniques like integration by parts or substitution. Lastly, finding limits of functions is a