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Continuity And Differentiability

Question
CBSEENMA12035315
Wired Faculty App

If space straight y equals cos to the power of negative 1 end exponent open parentheses fraction numerator 3 plus 5 space cos space straight x over denominator 5 plus 3 space cos space straight x end fraction close parentheses comma space prove space that space cos space straight x equals fraction numerator 4 minus 5 straight y subscript 1 over denominator 3 straight y subscript 1 end fraction comma space where space straight y subscript 1 equals dy over dx

Solution
space space space space space space space space space space space space straight y equals cos to the power of negative 1 end exponent open parentheses fraction numerator 3 plus 5 space cos space straight x over denominator 5 plus 3 space cos space straight x end fraction close parentheses space space therefore space dy over dx equals negative fraction numerator 1 over denominator square root of 1 minus open parentheses begin display style fraction numerator 3 plus 5 space cos space straight x over denominator 5 plus 3 space cos space straight x end fraction end style close parentheses squared end root end fraction. straight d over dx open parentheses fraction numerator 3 plus 5 space cos space straight x over denominator 5 plus 3 space cos space straight x end fraction close parentheses equals negative fraction numerator 5 plus 3 cos space straight x over denominator square root of left parenthesis 5 plus 3 cos space straight x right parenthesis squared minus left parenthesis 3 plus 5 space cos space straight x right parenthesis squared end root end fraction cross times fraction numerator left parenthesis 5 plus 3 cos space straight x right parenthesis left parenthesis negative 5 sin space straight x right parenthesis minus left parenthesis 3 plus 5 space cos space straight x right parenthesis left parenthesis negative 3 sin space straight x right parenthesis over denominator left parenthesis 5 plus 3 cos space straight x right parenthesis squared end fraction equals negative fraction numerator 1 over denominator square root of 25 plus 9 space cos squared space straight x minus 9 minus 25 space cos to the power of 2 space end exponent straight x end root end fraction cross times fraction numerator negative 25 space sin space straight x minus 15 space sin space straight x space cos space straight x plus 9 space sin space straight x plus 15 space sin space straight x space cos space straight x over denominator 5 plus 3 cos space straight x end fraction equals negative fraction numerator 1 over denominator square root of 16 minus 16 space cos squared space straight x end root end fraction cross times fraction numerator negative 16 space sin space straight x over denominator 5 plus 3 space cos space straight x end fraction equals negative fraction numerator 1 over denominator 4 space sin space straight x end fraction cross times fraction numerator negative 16 space sin space straight x over denominator 5 plus 3 space cos space straight x end fraction space space space space space therefore space straight y subscript 1 equals fraction numerator 4 over denominator 5 plus 3 space cos space straight x end fraction space space space straight R. straight H. straight S. equals fraction numerator 4 minus 5 straight y subscript 1 over denominator 3 straight y subscript 1 end fraction equals fraction numerator 4 minus begin display style fraction numerator 20 over denominator 5 plus 3 space cos space straight x end fraction end style over denominator begin display style fraction numerator 12 over denominator 5 plus 3 space cos space straight x end fraction end style end fraction equals fraction numerator 20 plus 12 space cos space straight x minus 20 over denominator 12 end fraction space space space space space space space space space space equals fraction numerator 12 space cos space straight x over denominator 12 end fraction equals cos space straight x equals straight L. straight H. straight S.

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