since
f(x) = 1 + 0.5*g(x)*g(x)
the derivative is
f'(x) = 0 + 2*0.5*g(x) * g'(x) = g(x)*g'(x)
note that the previous incorrect values do not affect the quality of
the obtained results (if the tolerance of the non-linear solver is
chosen to be small enough), but it may have deteriorated convergence
rates.
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