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Continuity And Differentiability

Question
CBSEENMA12034848
Wired Faculty App

Differentiate space square root of fraction numerator left parenthesis straight x minus 1 right parenthesis left parenthesis straight x minus 2 right parenthesis over denominator left parenthesis straight x minus 3 right parenthesis left parenthesis straight x minus 4 right parenthesis left parenthesis straight x minus 5 right parenthesis end fraction end root straight w. straight r. straight t. straight x

Solution
Let space space space space space space space straight y equals open square brackets fraction numerator left parenthesis straight x minus 1 right parenthesis left parenthesis straight x minus 2 right parenthesis over denominator left parenthesis straight x minus 3 right parenthesis left parenthesis straight x minus 4 right parenthesis left parenthesis straight x minus 5 right parenthesis end fraction close square brackets to the power of 1 half end exponent therefore space space log space straight y equals log open square brackets fraction numerator left parenthesis straight x minus 1 right parenthesis left parenthesis straight x minus 2 right parenthesis over denominator left parenthesis straight x minus 3 right parenthesis left parenthesis straight x minus 4 right parenthesis left parenthesis straight x minus 5 right parenthesis end fraction close square brackets to the power of 1 half end exponent space space space space space space space space space space space space space space space equals 1 half log open square brackets fraction numerator left parenthesis straight x minus 1 right parenthesis left parenthesis straight x minus 2 right parenthesis over denominator left parenthesis straight x minus 3 right parenthesis left parenthesis straight x minus 4 right parenthesis left parenthesis straight x minus 5 right parenthesis end fraction close square brackets space space space space space space space space space space space space space space space equals 1 half left square bracket log left curly bracket left parenthesis straight x minus 1 right parenthesis left parenthesis straight x minus 2 right parenthesis right curly bracket minus log left curly bracket left parenthesis straight x minus 3 right parenthesis left parenthesis straight x minus 4 right parenthesis left parenthesis straight x minus 5 right parenthesis right curly bracket right square bracket therefore space space log space straight y equals 1 half left square bracket log left parenthesis straight x minus 1 right parenthesis plus log left parenthesis straight x minus 2 right parenthesis plus log left parenthesis straight x minus 3 right parenthesis plus log left parenthesis straight x minus 4 right parenthesis plus log left parenthesis straight x minus 5 right parenthesis right square bracket Differentiating space both space sides space straight w. straight r. straight t. straight x comma space we space get comma space 1 over straight y dy over dx equals 1 half open square brackets fraction numerator 1 over denominator straight x minus 1 end fraction plus fraction numerator 1 over denominator straight x minus 2 end fraction minus fraction numerator 1 over denominator straight x minus 3 end fraction minus fraction numerator 1 over denominator straight x minus 4 end fraction minus fraction numerator 1 over denominator straight x minus 5 end fraction close square brackets therefore space space dy over dx equals 1 half straight y open square brackets fraction numerator 1 over denominator straight x minus 1 end fraction plus fraction numerator 1 over denominator straight x minus 2 end fraction minus fraction numerator 1 over denominator straight x minus 3 end fraction minus fraction numerator 1 over denominator straight x minus 4 end fraction minus fraction numerator 1 over denominator straight x minus 5 end fraction close square brackets therefore space space dy over dx equals 1 half square root of fraction numerator left parenthesis straight x minus 1 right parenthesis left parenthesis straight x minus 2 right parenthesis over denominator left parenthesis straight x minus 3 right parenthesis left parenthesis straight x minus 4 right parenthesis left parenthesis straight x minus 5 right parenthesis end fraction end root open square brackets fraction numerator 1 over denominator straight x minus 1 end fraction plus fraction numerator 1 over denominator straight x minus 2 end fraction minus fraction numerator 1 over denominator straight x minus 3 end fraction minus fraction numerator 1 over denominator straight x minus 4 end fraction minus fraction numerator 1 over denominator straight x minus 5 end fraction close square brackets

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