I am reading about linear stability analysis and have gotten stuck on some of the notation. For example, the book discusses finding the stability of the fixed points for x' = sinx. The book determines that f'(x) = cos(x). However, I thought that x' = sinx was equivalent to dx/dt = sinx where x is some function of t. Therefore wouldn't f'(x) be cosx * dx/dt = cosx*sinx? Where is my logic going wrong?
Thanks in advance.