Derivative of cosh x

The derivative of cosh x, [cosh x]',
is sinh x.

y = cosh² x means y = (cosh x)².

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

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

2(cosh x)¹ sinh x = 2 sinh x cosh x

So 2 sinh x cosh x is the derivative of cosh² x.