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Continuity And Differentiability

Question
CBSEENMA12035222
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Differentiate sin (tan-1 2x) w.r.t.x and express the result free from trigonometric symbols.

Solution
Let space space space space straight y equals sin left parenthesis tan to the power of negative 1 end exponent space 2 straight x right parenthesis Put space space 2 straight x equals tan space straight theta therefore space space space space space space straight y equals sin left parenthesis tan to the power of negative 1 end exponent left parenthesis tan space straight theta right parenthesis right parenthesis equals sin space straight theta therefore space space space space space space straight y equals fraction numerator 2 straight x over denominator square root of 1 plus 4 straight x squared end root end fraction therefore space dy over dx equals fraction numerator square root of 1 plus 4 straight x squared end root.2 minus 2 straight x. begin display style fraction numerator 8 straight x over denominator 2 square root of 1 plus 4 straight x squared end root end fraction end style over denominator 1 plus 4 straight x squared end fraction equals fraction numerator 2 square root of 1 plus 4 straight x squared end root minus begin display style fraction numerator 8 straight x squared over denominator square root of 1 plus 4 straight x squared end root end fraction end style over denominator 1 plus 4 straight x squared end fraction equals fraction numerator 2 left parenthesis 1 plus 4 straight x squared right parenthesis minus 8 straight x squared over denominator left parenthesis 1 plus 4 straight x squared right parenthesis to the power of begin display style 3 over 2 end style end exponent end fraction equals fraction numerator 2 over denominator left parenthesis 1 plus 4 straight x squared right parenthesis to the power of begin display style 3 over 2 end style end exponent end fraction

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