In calculus, the Leibniz notation for the second derivative confuses most students. It turns out, rightfully so. The notation for the second derivative turns out to actually be incorrect.
Earlier this year, I got a paper published detailing the problem and the corrected notation. In the video below, I introduce the new notation, why the old notation is problematic, how the new notation can be derived straightforwardly, and why it may matter in the future.
For those interested, the problems with the notation for the second derivative are well-known, and a kludge exists for working around it known as Faa di Bruno’s formula (a simplified version being known as “the chain rule for the second derivative”). However, this formula does not work out algebraically with the differentials. In the new notation, all of the differentials work together algebraically in a way that was not generally thought to be possible before.
For those interested, I have other (less radical) thoughts for improving the way that calculus is thought about and taught. You can also see them in action in my book Calculus from the Ground Up (shameless plug alert).