In the first panel, we have Newton's second law, then the statement that the force is minus the gradient of the gravitational potential V, followed by a calculation of the total energy in the system. The problem is to compute d/dt of the energy. (hint)
The answer is given in the last panel, below. According to the book, it's a "simple calculation."
The first step is to compute d/dt of the kinetic energy. We use the formula from above, plus a trick to convert the squared term back to the dot product of dr/dt with itself. Then we use the chain rule, and finally, the definition of the force in terms of the gradient of the potential.
Calculation of d/dt of the potential energy puzzled me for quite a while, though it really shouldn't have. My solution was to work backward from the answer, as shown.
We put the two results together, and notice that they cancel. Surprise!
