Let f(x)=x∣x∣, g(x)=sinx, and h(x)=(g∘f)(x). Then:
Solutions · 1
Official solution
h(x)={sin(x2),−sin(x2),x≥0x<0, h′(x)={2xcosx2,−2xcosx2,x≥0x<0, continuous at 0 (both sides →0).
h′′(x)={2cosx2−4x2sinx2,−2cosx2+4x2sinx2,x≥0x<0; the two one-sided limits at 0 are 2 and −2 --- not equal, so h′ is not differentiable at 0.