Re: Liebnitz notation in calculus
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Reply #19 –

Yes, it is a terrible tragedy that this never gets explained in textbooks! What is even more of a tragedy is that, I believe, the notation itself is incorrect. This is detectable if one actually takes

seriously as a differential.

To perform this operation, you would actually need to use the quotient rule. Doing so leads to a more expanded form of the second derivative:

Now, this reduces to

in the case of x being an independent variable. But, when x is not an independent variable, then the full expansion is needed.

If you have the full expansion, you can easily convert the second derivative of y with respect to x into the second derivative of x with respect to y using algebraic manipulations only, which is not possible with the traditional notation.