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 Dec 1 awarded Nice Answer Dec 26 accepted Mathematics Engineering: How do you prove the power rule? Sep 24 awarded Popular Question Sep 26 accepted Is $|x^r|=|x|^r$ for real numbers $x$ and $r$? Feb 9 awarded Teacher Dec 3 awarded Autobiographer 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 28 answered Mathematical equivalent of Feynman's Lectures on Physics? Oct 19 revised Derivative of polynomial function improved readability Oct 19 suggested approved edit on Derivative of polynomial function Oct 14 revised Proof of dividing fractional expressions improved readability Oct 12 revised Proof of dividing fractional expressions added 9 characters in body Oct 12 answered Proof of dividing fractional expressions 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 awarded Supporter 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 awarded Editor Oct 9 revised Mathematics Engineering: How do you prove the power rule? added 479 characters in body 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.