I went through engineering school, and 20 years of work (not as an engineer), before finding a calculus text that explained why the derivative of x^2 is 2x. Along with many practical applications of calculus.
That book was Calculus Made Simple, published in 1914. Thanks, Project Gutenberg!
Edit: derivative of x^2 is 2x. Got my differentiation and integration confused!
That’s exactly right. The proof is quite simple and there’s no reason it shouldn’t be taught instead of just getting students to accept magic rules.