You must have taken numerical methods a while ago! Newton isn’t guaranteed to converge, and the secant method has terrible worst-case performance. At least use Brent’s method, so you don’t lose the bracket on the root! (But most modern solvers use the ITP - Interpolate, Truncate and Project - algorithm of Oliveira and Takahashi.) Bisection is the easiest to code, and often Good Enough.
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