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Continuity And Differentiability

Question
CBSEENMA12034875
Wired Faculty App

If space straight y equals fraction numerator ax over denominator left parenthesis straight x minus straight a right parenthesis left parenthesis straight x minus straight b right parenthesis left parenthesis straight x minus straight c right parenthesis end fraction plus fraction numerator bx over denominator left parenthesis straight x minus straight b right parenthesis left parenthesis straight x minus straight c right parenthesis end fraction plus fraction numerator straight c over denominator straight x minus straight c end fraction plus 1. prove space that space fraction numerator straight y apostrophe over denominator straight y end fraction equals 1 over straight x open parentheses fraction numerator straight a over denominator straight a minus straight x end fraction plus fraction numerator straight b over denominator straight b minus straight x end fraction plus fraction numerator straight c over denominator straight c minus straight x end fraction close parentheses

Solution
space straight y equals fraction numerator ax squared over denominator left parenthesis straight x minus straight a right parenthesis left parenthesis straight x minus straight b right parenthesis left parenthesis straight x minus straight c right parenthesis end fraction plus fraction numerator bx over denominator left parenthesis straight x minus straight b right parenthesis left parenthesis straight x minus straight c right parenthesis end fraction plus fraction numerator straight c over denominator straight x minus straight c end fraction plus 1 space space space equals fraction numerator ax squared over denominator left parenthesis straight x minus straight a right parenthesis left parenthesis straight x minus straight b right parenthesis left parenthesis straight x minus straight c right parenthesis end fraction plus fraction numerator bx over denominator left parenthesis straight x minus straight b right parenthesis left parenthesis straight x minus straight c right parenthesis end fraction plus open parentheses fraction numerator straight c over denominator straight x minus straight c end fraction plus 1 close parentheses space space space equals fraction numerator ax squared over denominator left parenthesis straight x minus straight a right parenthesis left parenthesis straight x minus straight b right parenthesis left parenthesis straight x minus straight c right parenthesis end fraction plus fraction numerator bx over denominator left parenthesis straight x minus straight b right parenthesis left parenthesis straight x minus straight c right parenthesis end fraction plus fraction numerator straight x over denominator straight x minus straight c end fraction space space space equals fraction numerator ax squared over denominator left parenthesis straight x minus straight a right parenthesis left parenthesis straight x minus straight b right parenthesis left parenthesis straight x minus straight c right parenthesis end fraction plus fraction numerator bx plus straight x left parenthesis straight x minus straight b right parenthesis over denominator left parenthesis straight x minus straight b right parenthesis left parenthesis straight x minus straight c right parenthesis end fraction space space space equals fraction numerator ax squared over denominator left parenthesis straight x minus straight a right parenthesis left parenthesis straight x minus straight b right parenthesis left parenthesis straight x minus straight c right parenthesis end fraction plus fraction numerator straight x squared over denominator left parenthesis straight x minus straight b right parenthesis left parenthesis straight x minus straight c right parenthesis end fraction equals fraction numerator ax squared plus straight x squared left parenthesis straight x minus straight a right parenthesis over denominator left parenthesis straight x minus straight a right parenthesis left parenthesis straight x minus straight b right parenthesis left parenthesis straight x minus straight c right parenthesis end fraction therefore space straight y equals fraction numerator ax squared over denominator left parenthesis straight x minus straight a right parenthesis left parenthesis straight x minus straight b right parenthesis left parenthesis straight x minus straight c right parenthesis end fraction therefore space log space straight y equals log open square brackets fraction numerator straight x cubed over denominator left parenthesis straight x minus straight a right parenthesis left parenthesis straight x minus straight b right parenthesis left parenthesis straight x minus straight c right parenthesis end fraction close square brackets rightwards double arrow space log space straight y equals log space straight x cubed minus log left square bracket left parenthesis straight x minus straight a right parenthesis left parenthesis straight x minus straight b right parenthesis left parenthesis straight x minus straight c right parenthesis right square bracket therefore space log space straight y equals 3 log space straight x minus log left parenthesis straight x minus straight a right parenthesis minus log left parenthesis straight x minus straight b right parenthesis minus log left parenthesis straight x minus straight c right parenthesis Differentiating space straight w. straight r. straight t. straight x comma space we space get comma
space space space space space 1 over straight y dy over dx equals 3 over straight x minus fraction numerator 1 over denominator straight x minus straight a end fraction minus fraction numerator 1 over denominator straight x minus straight b end fraction minus fraction numerator 1 over denominator straight x minus straight c end fraction therefore space 1 over straight y dy over dx equals open parentheses 1 over straight x minus fraction numerator 1 over denominator straight x minus straight a end fraction close parentheses plus open parentheses 1 over straight x minus fraction numerator 1 over denominator straight x minus straight b end fraction close parentheses plus open parentheses 1 over straight x minus fraction numerator 1 over denominator straight x minus straight c end fraction close parentheses therefore space 1 over straight y dy over dx equals fraction numerator straight x minus straight a minus straight x over denominator straight x left parenthesis straight x minus straight a right parenthesis end fraction plus fraction numerator straight x minus straight b minus straight x over denominator straight x left parenthesis straight x minus straight b right parenthesis end fraction plus fraction numerator straight x minus straight c minus straight x over denominator straight x left parenthesis straight x minus straight c right parenthesis end fraction therefore space 1 over straight y dy over dx equals 1 over straight x open square brackets fraction numerator straight a over denominator straight a minus straight x end fraction plus fraction numerator straight b over denominator straight b minus straight x end fraction plus fraction numerator straight c over denominator straight c minus straight x end fraction close square brackets therefore space space space space space space space space fraction numerator straight y apostrophe over denominator straight y end fraction equals 1 over straight x open square brackets fraction numerator straight a over denominator straight a minus straight x end fraction plus fraction numerator straight b over denominator straight b minus straight x end fraction plus fraction numerator straight c over denominator straight c minus straight x end fraction close square brackets

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