The derivative of can be computed directly from this formula utilizing the binomial theorem, see for instance an alternative proof of the deriviative of (http://planetmath.org/AlternativeProofOfDerivativeOfXn).
However, to avoid invoking the binomial theorem one can often make use of alternative definitions of the derivative which are justified by inspecting a diagrams and/or through the use of algebra. Instead of using , use so that is the same as . This gives the formula
This is the standard slope formula between the two points and only now we let approach .
From this formula the casual rule
for positive integer values of can be easily derived.
First notice that
When is not a positive integer the proof typically depends on implicit differentiation as follows:
For the theoretically inclined this solution can be disappointing because it depends on a proof for the derivative of . Most texts simply redefine as the integral of or in some similar fashion delay an honest proof.
For this reason it is often instructive to prove the power rule in stages depending on the type of exponents. Having proven the result for a positive integer, one can extend this to using the product rule.
To begin with observe . Therefore
Now solve for .
Using the chain rule we can prove the power rule for as follows.
Once again solve for
Finally, the derivative of for any fraction is done by observing that so indeed the chain rule once again solves the problem.
|Date of creation||2013-03-22 15:50:10|
|Last modified on||2013-03-22 15:50:10|
|Last modified by||Algeboy (12884)|