Transcription of Thomas Calculus; 12th Edition: The Power Rule
1 Thomas calculus ; 12thEdition: The Power RuleClifford E. WeilSeptember 15, 2010 The following paragraph appears at the bottom of page 116 of ThomasCalculus, Power Rule is actually valid for all real numbersn. We have seenexamples for negative integers and fractional powers, butncould bean irrational number as well. to apply the Power Rule, we subtract1 from the original exponentnand multiply the result byn. Herewe state the general version of the rule, but postpone its proof untilChapter definition of the Power functionxaforaan irrational number isn tdefined until Chapter 7 after the introduction of the exponential function, it shouldn t be mentioned at this point in the text.
2 Establishing the PowerRule for functions of the formxrwhereris rational can be done using theChain Rule, which is covered in the current chapter. The first step is to provethe rule in the special casex1nwheren N. To do so, use the same techniqueused immediately preceding the paragraph mentioned above to verify the PowerRule forxn. Specifically forf(x) =x1nf (x) = limz xx1n z1nx z= limz xx1n z1n(x1n)n (z1n)n= limz xx1n z1n((x1n) (z1n))( n 1k=0(x1n)n 1 k(z1n)k)=1n(x1n)n 1=1nx1n 1 Next forn Napply the Chain Rule tox n= (x 1)nto extend the PowerRule toxnforn Z. Finally forr=mnwheren Nandm Zapply theChain Rule toxr= (x1n)mto prove the Power Ruleddxxr=rxr
