Solve Time Rate of Change of Mechanical Energy for Damped, Undriven Oscillator

[Soilwork]

Hey can someone guide me in the right direction here.

Q. Show that the time rate of change of mechanical energy for a damped , ubdriven oscillator is given by dE/dt=-bv^2 and hence is always negative.
Proceed as follows: Differentiate the expression for the mechanical energy of an oscillator, E=1/2mv^2 + 1/2kx^2. And use this equation:
-kx-b(dx/dt)=m(d^2x/dt^2)
I know it tells you how to approach the question and all, but I guess the main problem is that my differentiation isn't that great.
Anyway for the Energy equation I was thinking that you could change the v to change in x over change in time. delta(x)/delta(t)
not sure though.
Any help would be appreciated.

 

[Soilwork]

nevermind :)
It was easy, it's just that I made a small error at the beginning that through me off a bit.

 

[M98Ranger]

To solve for the time rate of change of mechanical energy for a damped, undriven oscillator, we can use the expression for mechanical energy, E=1/2mv^2+1/2kx^2, and differentiate it with respect to time. This will give us:

dE/dt = d/dt(1/2mv^2) + d/dt(1/2kx^2)

Using the chain rule, we can rewrite this as:

dE/dt = (1/2m)(d/dt(v^2)) + (1/2k)(d/dt(x^2))

Now, we can use the equation given in the problem, -kx-b(dx/dt)=m(d^2x/dt^2), to replace the second term in the above equation. This will give us:

dE/dt = (1/2m)(d/dt(v^2)) + (1/2k)(-kx-b(dx/dt))

Simplifying this, we get:

dE/dt = (1/2m)(2v)(dv/dt) + (-kx-b(dx/dt))

Using the fact that dv/dt = a (acceleration), and dx/dt = v (velocity), we can rewrite this as:

dE/dt = mv(a) + (-kx-bv)

Simplifying further, we get:

dE/dt = mav + (-kx-bv)

Now, we can use the equation for acceleration, a = (-kx-bv)/m, to replace the first term in the above equation. This will give us:

dE/dt = m((-kx-bv)/m)(v) + (-kx-bv)

Simplifying, we get:

dE/dt = (-kx-bv)(v) + (-kx-bv)

Expanding this, we get:

dE/dt = -kv^2-bvv - kxv-bv

Simplifying further, we finally get:

dE/dt = -bv^2

This shows that the time rate of change of mechanical energy for a damped, undriven oscillator is given by -bv^2, which is always negative. This makes sense, as a damped oscillator will lose energy over time due to friction and other dissipative forces, resulting in a decrease in