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

[nickadams]

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...




Thanks

 

[bp_psy]

You can prove the power rule from the limit definition of a derivative. Here is a link:
http://planetmath.org/encyclopedia/PowerRule2.html
Actually you can prove most derivative rules directly from the limit definition.

 

[Stephen Tashi]

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
$$(x + h)^n = x^n + n x^{n-1}h + ...$$.

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 $x^2$.

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!