Differential operators - the rules

[randybryan]

I always get slightly confused with the rules of differentials.

now \frac{d^{2}y}{dx^{2}} is the scond derivative of the function y(x

but rooting this does NOT give the first derivative dy/dx

However, with the operator \frac{d^{2}}{dx^{2}}, it seems that you can root this and it DOES give the first derivative.

Can someone please explain this to me? I may be wrong, but this seems to be the case in my quantum mechanic notes

 

[tiny-tim]

Hi randybryan!

Yes, d²y/dx² = (d/dx)(d/dx)y, but ≠ [(d/dx)y]².

It's for the same reason that sin(sin(y)) ≠ [sin(y)]²

 

[HallsofIvy]

I always get slightly confused with the rules of differentials.

now \frac{d^{2}y}{dx^{2}} is the scond derivative of the function y(x

but rooting this does NOT give the first derivative dy/dx

However, with the operator \frac{d^{2}}{dx^{2}}, it seems that you can root this and it DOES give the first derivative.

Can someone please explain this to me? I may be wrong, but this seems to be the case in my quantum mechanic notes

I suspect that your "quantum mechanic notes" (don't hold mathematicians responsible for what a physicist says!) are using a special notation in which "square root" has some kind of operational definition. That is, "f^2(x)" does not mean f(x) times itself but f(f(x)) and \sqrt{f} is the inverse of that.

 

[randybryan]

Ahh the ongoing feud that is Maths nomenclature vs Physics nomenclature.