Limits and Derivatives

Limits and Derivatives

Question

Derivative of (ax + b)n, n being a rational number.

Answer

Let           f(x) = (ax+b)n

 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 open square brackets straight a open parentheses straight x plus straight h close parentheses plus straight b close square brackets to the power of straight n minus left parenthesis ax plus straight b 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 space space equals space limit as straight h rightwards arrow 0 of fraction numerator left square bracket ax plus straight b plus ha right square bracket to the power of straight n minus left parenthesis ax plus straight b 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 equals space limit as straight h rightwards arrow 0 of left parenthesis ax plus straight b right parenthesis to the power of straight n fraction numerator open curly brackets open parentheses 1 plus begin display style fraction numerator ah over denominator ax plus straight b end fraction end style close parentheses to the power of straight n minus 1 close curly brackets over denominator straight h end fraction 

    space space space space space space space space space space space space space space space space space space space space space space equals space left parenthesis a x plus b right parenthesis to the power of n. limit as h rightwards arrow 0 of 1 over h open curly brackets open parentheses 1 plus fraction numerator n h a over denominator a x plus b end fraction plus fraction numerator n left parenthesis n minus 1 right parenthesis over denominator left enclose bottom enclose 2 end enclose end fraction. fraction numerator h squared a squared over denominator left parenthesis a x plus b right parenthesis squared end fraction plus..... close parentheses minus 1 close curly brackets

         space space space space space space space space space space space space space space space space space space space space space space equals space left parenthesis ax plus straight b right parenthesis to the power of straight n. limit as straight h rightwards arrow 0 of open curly brackets fraction numerator nha over denominator ax plus straight b end fraction plus fraction numerator straight n left parenthesis straight n minus 1 right parenthesis over denominator bottom enclose left enclose 2 end enclose end fraction. fraction numerator straight h squared straight a squared over denominator left parenthesis ax plus straight b right parenthesis squared end fraction plus...... close curly brackets

      space space space space space space space space space space space space equals space left parenthesis ax plus straight b right parenthesis to the power of straight n. limit as straight h rightwards arrow 0 of 1 over straight h open curly brackets fraction numerator na over denominator ax plus straight b end fraction plus fraction numerator straight n left parenthesis straight n minus 1 right parenthesis over denominator bottom enclose left enclose 2 end enclose end fraction. fraction numerator ha squared over denominator left parenthesis ax plus straight b right parenthesis squared end fraction plus...... close curly brackets
     space space space space space space space space equals space left parenthesis ax plus straight b right parenthesis to the power of straight n open parentheses fraction numerator na over denominator ax plus straight b end fraction plus 0 close parentheses equals na left parenthesis ax plus straight b right parenthesis to the power of straight n minus 1 end exponent
Hence,  space space space space space space straight d over dx left parenthesis ax plus straight b right parenthesis to the power of straight n equals space na left parenthesis ax plus straight b right parenthesis to the power of straight n minus 1 end exponent
Corollary :  Put a = 1m, b = 0

space space 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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