f(t)=(t^3, |t|^3) is a parametric representation of y=f(x)=|x|. Consider y=|x|, the left hand derivative f '-(0)=-1 and the right hand derivative f '+(0)=1, so f(x) is clearly not differentiable at 0. But f '(t)=(3t^2, 3t^2) for t>=0 f '(t)=(3t^2, -3t^2) for t<=0 f '(0)=(0,0) and f(t) is differentiable at 0 (my textbook says this explicitly) These 2 are talking about the same point, how can one gives that it's differentiable at 0 and the other gives that it's not differentiable at 0? ============================================== Secondly, my textbook says that f '(a)>0 (a is real number) doesn't imply that f is increasing in some neighbourhood of a, how come? Thanks for explaining!