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- TL;DR Summary
- About the mass-spring system.

Can we derive one equation of motion from another?

In chapter 3 of Vibrations and Waves by French, there is a description about the equations of motion of a mass-spring system. It was written as shown in the attached picture:

Here, m is the mass on the spring, k is the spring constant, x is the extension of the spring, and t is time. My problem is about figuring out how equation (3-1) becomes equation (3-2)? I tried to differentiate 1/2 m(dx/dt)2:

I got an extra dt/dx which is not there in equation (3-1). Is my working wrong or is there another way of understanding what is in the textbook: "The second is, of course, the result of integrating the first with respect to the displacement x ..."?

Here, m is the mass on the spring, k is the spring constant, x is the extension of the spring, and t is time. My problem is about figuring out how equation (3-1) becomes equation (3-2)? I tried to differentiate 1/2 m(dx/dt)2:

I got an extra dt/dx which is not there in equation (3-1). Is my working wrong or is there another way of understanding what is in the textbook: "The second is, of course, the result of integrating the first with respect to the displacement x ..."?