My dad (an engineer not a mathematician) would use Newton-Raphson[1] to solve basically any problem that wasn’t very obviously linear. When I was a kid, some of my first programming memories were my dad getting me and my brother to implement Newton-Raphson in basic on an HP85a, getting me to implement Newton-Raphson in rpn on an HP calculator, debugging my dad’s (genuinely revolting) basic program[2] which wouldn’t run but (who would have guessed?) used Newton-Raphson to compute something or other.

He basically learned the one numerical root-finder and how to evaluate basic second derivatives and he was set for life on all the problems a career in chemical and process engineering could throw at someone.

[1] https://sheffield.ac.uk/media/31988/download?attachment

[2] He learned to program in FORTRAN and lived by the maxim that a determined FORTRAN programmer can write FORTRAN in any language.