Lecture 2. B) Leibniz Rule II

An Approach to Understanding Leibniz Rule with Improper Integrals

Here is a related explanation of why Leibniz Rule fails under the improper integral of [math]f\left(x,t\right)=\frac{\sin\left(tx\right)}{t}[/math].

Consider the plot below, where [math]x[/math] varies between 1 and 4.

As [math]x[/math] increases, the graph is squeezed in the horizontal direction, in the same amount that it is stretched vertically. This is the reason the 'area under the graph' remains constant over [math]x[/math].

However, if we consider any specific value of [math]t[/math], we notice that the area under the curve surrounding is periodic as [math]x[/math] changes.

Importantly, this function remains periodic for all values of [math]t[/math], so that the sum of such area changes does not converge.

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(Thanks to Matthias Goerner for providing this intuition.)