Limits and Derivatives

Limits and Derivatives

Question

Derivative of xn.

Answer

Let          f(x) = xn
                   
space space space space space space space space space space space space space space space space space space space space space space space space space straight f left parenthesis straight x right parenthesis space equals space limit as straight h rightwards arrow 0 of fraction numerator straight f left parenthesis straight x plus straight h right parenthesis minus straight f left parenthesis straight x right parenthesis over denominator straight h end fraction equals limit as straight h rightwards arrow 0 of fraction numerator left parenthesis straight x plus straight h right parenthesis to the power of straight n minus left parenthesis straight x right parenthesis to the power of straight n over denominator straight h end fraction
 
              space space space space space space space space space space space space space space space equals space space limit as straight h rightwards arrow 0 of fraction numerator open square brackets straight x open parentheses 1 plus begin display style straight h over straight x end style close parentheses close square brackets to the power of straight n minus straight x to the power of straight n over denominator straight h end fraction equals limit as straight h rightwards arrow 0 of fraction numerator straight x to the power of straight n open parentheses 1 plus begin display style straight h over straight x end style close parentheses to the power of straight n minus straight x to the power of straight n over denominator straight h end fraction
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                     space space space space space space space space space space equals space space space straight x to the power of straight n space space limit as straight h rightwards arrow 0 of fraction numerator open parentheses 1 plus begin display style nh over straight x end style plus begin display style fraction numerator straight n left parenthesis straight n minus 1 right parenthesis over denominator 2 factorial end fraction end style plus begin display style h squared over x squared end style plus.... close parentheses minus 1 over denominator straight h end fraction

                       space space space space space space space space space equals space straight x to the power of straight n limit as straight h rightwards arrow 0 of fraction numerator straight h open square brackets begin display style straight n over straight x end style plus begin display style fraction numerator straight n left parenthesis straight n minus 1 right parenthesis over denominator 21 end fraction end style begin display style straight h over straight x squared end style plus..... close square brackets over denominator straight h end fraction

        space space space space space space space space space space space space space space space space equals space space straight x to the power of straight n space limit as straight h rightwards arrow 0 of open square brackets straight n over straight x plus fraction numerator straight n left parenthesis straight n minus 1 right parenthesis over denominator 2 factorial end fraction space straight h over straight x squared plus.... close square brackets equals straight x to the power of straight n. end exponent straight n over straight x equals nx to the power of straight n minus 1 end exponent

       Hence,   space space space space straight d over dx left parenthesis straight x to the power of straight n right parenthesis space equals space nx to the power of straight n minus 1 end exponent

              

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