Let f(x)=xx|=xx|,g(x)=sinx and h(x)=gof(x)=g[f(x)] ∴h(x)={
sinx2,
x≥0
−sinx2,
x<0
Now, h′(x)={
2xcosx2,
x≥0
−2xcosx2,
x<0
Since, L.H.L and R.H.L at x = 0 of h'(x) is equal to 0 therefore h'(x) is continuous at x = 0 Now, suppose h'(x) is differentiable
∴h"(x)={
2(cosx2+2x2(−sinx2),
x≥0
2(−cosx2+2x2sinx2),
x<0
Since, L.H.L and R.H.L at x = 0 of h"(x) are different therefore h"(x) is not continuous. ⇒ h"(x) is not differentiable ⇒ our assumption is wrong Hence h'(x) is not differentiable at x = 0.