Is the derivative in my textbook correct here?

[whitejac]

Homework Statement

Homework Equations

d/dx

The Attempt at a Solution

d/dx (T) = d/dx(1/2mx'²) = mx''
d/dx(U) = d/dx(1/2kx²) = kx' ≠ kx

It's probably me who made an error because I know that that equation (2.3) is the one I should be getting, but I don't understand how they did it because potential energy relies on position, so the derivative has to be a velocity vector.

 

[gneill]

Here x is a function of t and the derivatives should be with respect to t. That's what x' implies.

 

[NascentOxygen]

(2.6) shows differentiation with respect to time. Your working is erroneously using differentiation with respect to x.

 

[whitejac]

Oh right, it's x(t)... I guess the caffeine hasn't kicked in because that was really basic. Thank you!

 

[gneill]

No worries. Enjoy your coffee!

 

[NascentOxygen]

I think (2.3) starts out as $m \ddot x \dot x + k x \dot x = 0$ and the common factor $\dot x$ can be canceled from both sides.