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Why does the power rule work? [answer is probably :/ obvious]

  1. Aug 28, 2011 #1
    Can someone explain to me in intuitive terms why the trick of bringing the the exponent out front and then reducing the power by 1 works?

    Solving problems using the Limit definition of a derivative (where we take a secant line closer and closer to a point) makes great intuitive sense, but I can't wrap my feeble mind around where the shortcut method for finding a derivative comes from...

    Last edited: Aug 28, 2011
  2. jcsd
  3. Aug 28, 2011 #2
    You can prove the power rule from the limit definition of a derivative. Here is a link:
    http://planetmath.org/encyclopedia/PowerRule2.html [Broken]
    Actually you can prove most derivative rules directly from the limit definition.
    Last edited by a moderator: May 5, 2017
  4. Aug 29, 2011 #3

    Stephen Tashi

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    As by_psy suggests, if you have an intutition for the manipulation of symbols, you can understand that the power rule for positive integer exponents N stems from the fact that
    [tex] (x + h)^n = x^n + n x^{n-1}h + .... [/tex].

    If you only have intuition for geometry, you can understand it some simple cases. For example, draw a square with sides of length x and area x^2, leave one vertex fixed while increasing the lengths of the sides to x+h. You can see that there are two big strips of additional of size (x+h) by h and one small square of area that is h by h. Having 2 strips is a consequence of being in two dimensions and is related to the exponent 2 in the area [itex] x^2 [/itex].

    As for having intuition about the cases where n is negative, fractional or irrational - the only intuition I have about that is the "power of suggestion" from knowing the result when n is a positive integer!
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