Let i cr be an interval and f, 9: 1 + r be differentiable functions at cei. by caratheodory's theorem there exist functions 6,0: 1 + r which are continuous at c, with 6 (c) = f'(c) and 4 (c) = g' (c) such that for all xe i we have f (x) = f (c) + 6(x) (x – c), g(x) = g(e) + (l) (x – c) use this criteria to prove: 1. the product rule of differentiation; 2. the differentiation formula for inverse functions.