Derivative of sin x

x sin x
is the product of x and sin x.

So the derivative of the product is,
the derivative of x, 1
times sin x
plus
x
times, the derivative of sin x, cos x.

1⋅sin x = sin x

So
sin x + x cos x
is the derivative of x sin x.

y = sin x²
is a composite function of
y = sin (whole) and (whole) = x².

So the derivative of the composite function is,
the derivative of the outer function sin x², cos x²
times,
the derivative of the inner function x², 2x¹.

(cos x²)⋅(2x¹) = 2x cos x²

So
2x cos x²
is the derivative of sin x².

y = sin² x
means
y = (sin x)².

This is a composite function of
y = (whole)² and (whole) = sin x.

So the derivative of the composite function is,
the derivative of the outer function (sin x)², 2(sin x)¹
times,
the derivative of the inner function sin x, cos x.

2(sin x)¹⋅cos x = 2 sin x cos x

Then, by the sin 2A formula,
2 sin x cos x = sin 2x.

So
sin 2x
is the derivative of sin² x.