find the derivate of x raise to n from first principle?

Asked by raghavirao12 | 21st Nov, 2014, 09:36: PM

Expert Answer:

G i v e n space t h a t space f open parentheses x close parentheses equals x to the power of n W e space n e e d space t o space d i f f e r e n t i a t e space t h e space g i v e n space f u n c t i o n space f r o m space t h e space f i r s t space p r i n c i p l e. T h e r e f o r e comma space w e space h a v e comma fraction numerator d y over denominator d x end fraction equals limit as h rightwards arrow 0 of fraction numerator f open parentheses x plus h close parentheses minus f open parentheses x close parentheses over denominator h end fraction rightwards double arrow fraction numerator d y over denominator d x end fraction equals limit as h rightwards arrow 0 of fraction numerator open parentheses x plus h close parentheses to the power of n minus x to the power of n over denominator h end fraction F r o m space B i n o m i a l space T h e o r e m comma space open parentheses x plus h close parentheses to the power of n equals x to the power of n plus to the power of n C subscript 1 x to the power of n minus 1 end exponent cross times h plus to the power of n C subscript 2 x to the power of n minus 1 end exponent cross times h squared plus to the power of n C subscript 3 x to the power of n minus 2 end exponent cross times h cubed plus... plus to the power of n C subscript n minus 1 end subscript x to the power of 1 cross times h to the power of n minus 1 end exponent plus to the power of n C subscript n x to the power of 0 cross times h to the power of n T h u s comma space w e space h a v e comma fraction numerator d y over denominator d x end fraction equals limit as h rightwards arrow 0 of open parentheses fraction numerator x to the power of n plus to the power of n C subscript 1 x to the power of n minus 1 end exponent cross times h plus to the power of n C subscript 2 x to the power of n minus 1 end exponent cross times h squared plus to the power of n C subscript 3 x to the power of n minus 2 end exponent cross times h cubed plus... plus to the power of n C subscript n minus 1 end subscript x to the power of 1 cross times h to the power of n minus 1 end exponent plus to the power of n C subscript n x to the power of 0 cross times h to the power of n minus x to the power of n over denominator h end fraction close parentheses rightwards double arrow fraction numerator d y over denominator d x end fraction equals limit as h rightwards arrow 0 of open parentheses fraction numerator blank to the power of n C subscript 1 x to the power of n minus 1 end exponent cross times h plus to the power of n C subscript 2 x to the power of n minus 1 end exponent cross times h squared plus to the power of n C subscript 3 x to the power of n minus 2 end exponent cross times h cubed plus... plus to the power of n C subscript n minus 1 end subscript x to the power of 1 cross times h to the power of n minus 1 end exponent plus h to the power of n over denominator h end fraction close parentheses rightwards double arrow fraction numerator d y over denominator d x end fraction equals limit as h rightwards arrow 0 of open parentheses n x to the power of n minus 1 end exponent plus to the power of n C subscript 2 x to the power of n minus 1 end exponent h plus to the power of n C subscript 3 x to the power of n minus 2 end exponent cross times h squared plus... plus to the power of n C subscript n minus 1 end subscript x to the power of 1 cross times h to the power of n minus 2 end exponent plus h to the power of n minus 1 end exponent close parentheses rightwards double arrow fraction numerator d y over denominator d x end fraction equals n x to the power of n minus 1 end exponent space space space space space space left square bracket limit as h rightwards arrow 0 of open parentheses blank to the power of n C subscript 2 x to the power of n minus 1 end exponent h plus to the power of n C subscript 3 x to the power of n minus 2 end exponent cross times h squared plus... plus to the power of n C subscript n minus 1 end subscript x to the power of 1 cross times h to the power of n minus 2 end exponent plus h to the power of n minus 1 end exponent close parentheses equals 0 right square bracket

Answered by Vimala Ramamurthy | 22nd Nov, 2014, 08:14: AM

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