2nd derivative for Acceleration

[REcr]

I came across this derivation for acceleration 1/2(d/dx)(dx/dt)^2=d²x/dt² , this cant be correct can it? if so can you point me in the direction of a proof for it.

 

[Steven G]

I came across this derivation for acceleration 1/2(d/dx)(dx/dt)^2=d²x/dt² , this cant be correct can it? if so can you point me in the direction of a proof for it.

For a moment let f(x)=dx/dt.
So we want to simplify (1/2)(d/dx)[f(x)^2] =f(x)*f'(x) by the chain rule. Well f(x)=dx/dt and f'(x) =d²x/dt² , so their product will not be d²x/dt².


EDIT: oops you had x's and t's! I did not see that. I really must wear my eye glasses all the time.

 

[Deleted member 4993]

I came across this derivation for acceleration 1/2(d/dx)(dx/dt)^2=d²x/dt² , this cant be correct can it? if so can you point me in the direction of a proof for it.

a = dv/dt = dv/dx * dx/dt = dv/dx * v = d/dx (1/2 v²) = 1/2 (d/dx v²) = 1/2 d/dx (dx/dt)²

This is how the Galileo's third equation (v² = u² + 2*a*s) was confirmed theoretically.