## 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

Then

## Special cases

In fact, these special cases allow one to see how the Leibniz rule is obtained.

### Case 1

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

### Case 2

depends only on

Then

(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.

### Case 3

depends on .

Then

(only the integrand is differentiated).

**Intuition**.

which leads to differentiation under the integral sign.

### Putting it all together

In the general case we can denote

so that

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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