Compare the derivatives for y=sin^2(x) And y=sin(x^2)?

When y = sin²(x):
dy/dx = [d sin²(x)]/dx
= {[d sin²(x)]/d sin(x)} * [d sin(x) / dx]
= 2 sin(x) * cos(x)
= 2 sin(x) cos(x)
= sin(2x)

When y = sin(x²):
dy/dx = [d sin(x²)] / dx
= {[d sin(x²)] / d(x²)} * [d (x²)/dx]
= cos(x²) * 2 x
= 2 x cos(x²)

y = sin^2x
dy/dx = 2sinx cosx
y = sin(x^2)
dy/dx = 2xsin(x^2)

y1 = sin^2(x), y1' = 2sinx*cosx  = sin(2x).
y2 = sin(x^2), y2' = cos(x^2)*2x = 2xcos(x^2).

y'= sin(2x) and y'= 2xcos(x²)