Sony
Reputation
Next privilege 250 Rep.
 Dec26 accepted Mathematics Engineering: How do you prove the power rule? Sep24 awarded Popular Question Sep26 accepted Is $|x^r|=|x|^r$ for real numbers $x$ and $r$? Feb9 awarded Teacher Dec3 awarded Autobiographer Nov16 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. Oct28 answered Mathematical equivalent of Feynman's Lectures on Physics? Oct19 revised Derivative of polynomial function improved readability Oct19 suggested approved edit on Derivative of polynomial function Oct14 revised Proof of dividing fractional expressions improved readability Oct12 revised Proof of dividing fractional expressions added 9 characters in body Oct12 answered Proof of dividing fractional expressions Oct9 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. Oct9 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? Oct9 awarded Supporter Oct9 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. Oct9 awarded Editor Oct9 revised Mathematics Engineering: How do you prove the power rule? added 479 characters in body Oct9 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. Oct9 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?