Assume that f and g are differentiable functions defined on all of the real line. f and g can satisfy: for all x and for all x.

Question

Assume that f and g are differentiable functions defined on all of the real line. f and g can satisfy:  for all x and  for all x.

Quizplus · Accepted Answer

This statement is true by the product rule of differentiation, which says that (fg)' = f'g + fg'. Since the statement given in the question says that (fg)' = f'g + fg, it follows that f'g = gf', which is satisfied by functions of the form f(x) = Ae^kx and g(x) = Be^kx, where A and B are constants and k is a constant that is the same for both functions.

Assume That F and G Are Differentiable Functions Defined on All