Could you define the RHS a bit more in detail? I am having trouble understanding it.I think you are getting hung up on notation. I interpret the left hand side as "the derivative of the function P(t)". I interpret the right hand side as "P prime of t with respect to t".
[math]\frac{d(\phi(t))}{dt}=\phi'(t)\\ d(\phi(t))=\phi'(t)dt[/math]Could you define the RHS a bit more in detail? I am having trouble understanding it.