Suppose fx x2 and gx x Then the composites fogx x2 x2 an

Suppose f(x) = x^2 and g(x) = |x|. Then the composites (fog)(x) = |x|^2 = x^2 and (gof)(x) = |x^2| = x^2 are both differentiable at x = 0 even though g itself is not differentiable at x = 0. Does this contradict the chain rule? Explain.

Solution

It does not because the chain rule stipulates that both f and g are differentiable at x = c = 0. It doesn\'t conclude anything if g is not differentiable in the first place. This relates to the fact that \"if p then q\" is not logically equivalent to \"if q then p\" and that if the hypothesis p is false, then the statement is vacously true.