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Continuity And Differentiability

Question
CBSEENMA12034879
Wired Faculty App

Differentiate (x2-5x+8)(x3+7x+ 9) in three ways mentioned below :
(i) by using product rule.
(ii) by expanding the product to obtain a single polynomial.
(iii) by logarithmic differentiation.
Also verify that three answers so obtained are the same.

Solution
Let space space space space space space space space space straight y equals left parenthesis straight x squared minus 5 space straight x plus 8 right parenthesis left parenthesis straight x cubed plus 7 straight x plus 9 right parenthesis left parenthesis straight i right parenthesis therefore space dy over dx equals left parenthesis straight x squared minus 5 space straight x plus 8 right parenthesis. straight d over dx left parenthesis straight x cubed plus 7 straight x plus 9 right parenthesis plus left parenthesis straight x cubed plus 7 straight x plus 9 right parenthesis. straight d over dx left parenthesis straight x squared minus 5 space straight x plus 8 right parenthesis space space space space space space equals left parenthesis straight x squared minus 5 space straight x plus 8 right parenthesis left parenthesis 3 straight x squared plus 7 right parenthesis plus left parenthesis straight x cubed plus 7 straight x plus 9 right parenthesis left parenthesis 2 straight x minus 5 right parenthesis space space space space space space equals 3 straight x to the power of 4 plus 7 straight x squared minus 15 straight x cubed minus 35 straight x plus 24 straight x squared plus 56 plus 2 straight x to the power of 4 minus 5 straight x cubed plus 14 straight x squared minus 35 straight x plus 18 straight x minus 45 space space space space space space equals 5 straight x to the power of 4 minus 20 straight x cubed plus 45 straight x squared minus 5 straight x minus 52 straight x plus 11 left parenthesis ii right parenthesis space space space space space space space space space space space straight y equals straight x to the power of 5 minus 5 straight x to the power of 4 plus 15 straight x cubed minus 26 straight x squared plus 11 straight x plus 72 therefore 1 over straight y dy over dx equals 5 straight x to the power of 4 minus 20 straight x 3 plus 45 straight x squared minus 52 straight x plus 11 left parenthesis iii right parenthesis space space space space log space straight y equals log left parenthesis straight x squared minus 5 space straight x plus 8 right parenthesis plus log left parenthesis straight x cubed plus 7 straight x plus 9 right parenthesis therefore space 1 over straight y dy over dx equals fraction numerator 2 straight x minus 5 over denominator straight x squared minus 5 space straight x plus 8 end fraction plus fraction numerator 3 straight x squared plus 7 over denominator straight x cubed plus 7 straight x plus 9 end fraction therefore space space space space space space space dy over dx equals left parenthesis straight x squared minus 5 space straight x plus 8 right parenthesis left parenthesis straight x cubed plus 7 straight x plus 9 right parenthesis open parentheses fraction numerator 2 straight x minus 5 over denominator straight x squared minus 5 space straight x plus 8 end fraction plus fraction numerator 3 straight x squared plus 7 over denominator straight x cubed plus 7 straight x plus 9 end fraction close parentheses space space space space space space space space space space space space space space space space space space space equals left parenthesis 2 straight x minus 5 right parenthesis left parenthesis straight x cubed plus 7 straight x plus 9 right parenthesis plus left parenthesis 3 straight x squared plus 7 right parenthesis left parenthesis straight x squared minus 5 space straight x plus 8 right parenthesis space space space space space space space space space space space space space space space space space space space equals 5 straight x to the power of 4 minus 20 straight x cubed plus 45 straight x squared minus 52 straight x plus 11 Answer space in space the space all space the space cases space is space same.

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