Leibniz integral rule
This rule is about differentiating an integral that has a parameter in three places: the lower and upper limits of integration and in the integrand.
Let be the lower and upper limits of integration and
In fact, these special cases allow one to see how the Leibniz rule is obtained.
depends only on Then
(the upper limit goes into the argument of ).
Intuition. By definition, is the limit of when (From Chart 1, in which , it is seen that this ratio is almost a slope at point .) Hence,
is the limit of
When this approximation becomes better and better and in the limit
depends only on
(the sign is opposite to Case 1).
Intuition. One of the properties of integral is that
To the last integral we can apply the intuition for Case 1.
depends on .
(only the integrand is differentiated).
which leads to differentiation under the integral sign.
Putting it all together
In the general case we can denote
By the chain rule
giving us the general Leibniz rule.
Note that Case 1 implies
(the density is found from the distribution function).
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