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 Nov 16 comment Mathematical equivalent of Feynman's Lectures on Physics? No votes for Richard Courant? Do I have to say more about this book? Now I am puzzled. Oct 9 comment Mathematics Engineering: How do you prove the power rule? Thank you very much. You confirmed my suspicion. You cannot prove power rule in Grade 12 if you want to be coherent. Hung-Hsi Wu in his book "Understanding Numbers in Elementary School Mathematics" introduces the term FASM (Fundamental Assumption of School Mathematics). That is, "All information about the arithmetic operations on fractions can be extrapolated to all real numbers". We may have to use something similar to FASM to get the general power rule in Grade 12. Better yet, stop (the proof) at rational powers and do not do any problem involving irrational powers. Oct 9 comment Mathematics Engineering: How do you prove the power rule? I followed your proof up to rational powers. In the real case, you use "exp function is differentiable" and "ln function is differentiable" How do you establish one or the other at Grade 12 level? Oct 9 comment Mathematics Engineering: How do you prove the power rule? @ArthroMagidin: I like your proof. Let us assume that the hypothetical Grade 12 student has a very good understanding what limit of a function (at a point) is. Oct 9 comment Mathematics Engineering: How do you prove the power rule? You use $\ln x$ in $e^{r\ln X}$. How do you know $x>0$? By definition, (Grade 12 level) a real-valued function is differentiable at $x_0$ if the derivative exists at $x_0$. Therefore, existence of the derivative must be established first. Hence, the question 1. Oct 9 comment Mathematics Engineering: How do you prove the power rule? To apply log (I presume log base 10) $y$ has to be positive. How do you know that $y$ is positive? Oct 9 comment Mathematics Engineering: How do you prove the power rule? @RossMillikan: I just looked at it. The generalization part is vague. Sep 23 comment Is $|x^r|=|x|^r$ for real numbers $x$ and $r$? Thank you to both Stefan Geschke and Shaun Ault.