Derivative of csc x

The derivative of csc x, [csc x]',
is -csc x cot x.

y = csc (1 - x²) is a composite function of
y = csc (whole) and (whole) = 1 - x².

So the derivative of the composite function is,
the derivative of the outer function csc (1 - x²), -csc (1 - x²) cot (1 - x²)
times,
the derivative of the inner function 1 - x², -2x¹.

-(-2x¹) = 2x

So the right side is 2x csc (1 - x²) cot (1 - x²).

So 2x csc (1 - x²) cot (1 - x²) is the derivative of csc (1 - x²).