Question: The derivative f'gives the (absolute) rate of change of a quantity f, and f'/f gives the relative rate of change of the quantity. In this problem, we show that the product rule is equivalent to an additive rule for relative rates of change. Assume h = f · g with f ≠ 0 and g ≠ 0.
(a) Show that the additive rule
f'/f + g'/g = h'/h
implies the product rule, by multiplying through by h and using the fact that h = f · g.
(b) Show that the product rule implies the additive rule in part (a), by starting with the product rule and dividing through by h = f · g.