D i f f e r e n t i a t e space tan to the power of minus 1 end exponent space space open parentheses fraction numerator square root of 1 plus a squared x squared end root space minus 1 over denominator a x end fraction close parentheses space w. r. t. x.

Asked by inderjeet.saluja | 24th Aug, 2014, 05:42: PM

Expert Answer:

l e t space f left parenthesis x right parenthesis equals tan space to the power of minus 1 end exponent open parentheses fraction numerator square root of 1 plus a squared x squared end root minus 1 over denominator a x end fraction close parentheses f to the power of apostrophe left parenthesis x right parenthesis equals fraction numerator 1 over denominator 1 plus open parentheses fraction numerator square root of 1 plus a squared x squared end root minus 1 over denominator a x end fraction close parentheses squared end fraction cross times fraction numerator d over denominator d x end fraction open square brackets fraction numerator square root of 1 plus a squared x squared end root minus 1 over denominator a x end fraction close square brackets equals fraction numerator a squared x squared over denominator a squared x squared plus open parentheses square root of 1 plus a squared x squared end root minus 1 close parentheses squared end fraction cross times 1 over a open square brackets fraction numerator x cross times begin display style fraction numerator 2 a squared x over denominator 2 square root of 1 plus a squared x squared end root end fraction end style minus open parentheses square root of 1 plus a squared x squared end root minus 1 close parentheses cross times 1 over denominator x squared end fraction close square brackets equals fraction numerator a x squared over denominator a squared x squared plus open parentheses square root of 1 plus a squared x squared end root minus 1 close parentheses squared end fraction cross times open square brackets fraction numerator begin display style fraction numerator a squared x squared over denominator square root of 1 plus a squared x squared end root end fraction end style minus open parentheses square root of 1 plus a squared x squared end root minus 1 close parentheses cross times 1 over denominator x squared end fraction close square brackets equals fraction numerator a x squared over denominator a squared x squared plus open parentheses square root of 1 plus a squared x squared end root minus 1 close parentheses squared end fraction cross times open square brackets fraction numerator a squared x squared minus open parentheses square root of 1 plus a squared x squared end root close parentheses squared plus square root of 1 plus a squared x squared end root over denominator x squared open parentheses square root of 1 plus a squared x squared end root close parentheses end fraction close square brackets equals fraction numerator a x squared over denominator a squared x squared plus open parentheses square root of 1 plus a squared x squared end root minus 1 close parentheses squared end fraction cross times open square brackets fraction numerator a squared x squared minus 1 minus a squared x squared plus square root of 1 plus a squared x squared end root over denominator x squared open parentheses square root of 1 plus a squared x squared end root close parentheses end fraction close square brackets equals fraction numerator a x squared over denominator a squared x squared plus open parentheses square root of 1 plus a squared x squared end root minus 1 close parentheses squared end fraction cross times open square brackets fraction numerator square root of 1 plus a squared x squared end root minus 1 over denominator x squared open parentheses square root of 1 plus a squared x squared end root close parentheses end fraction close square brackets equals fraction numerator a open parentheses square root of 1 plus a squared x squared end root minus 1 close parentheses over denominator a squared x squared plus open parentheses square root of 1 plus a squared x squared end root minus 1 close parentheses squared end fraction cross times fraction numerator 1 over denominator open parentheses square root of 1 plus a squared x squared end root close parentheses end fraction equals fraction numerator a open parentheses square root of 1 plus a squared x squared end root minus 1 close parentheses over denominator a squared x squared plus 1 plus a squared x squared plus 1 minus 2 square root of 1 plus a squared x squared end root end fraction cross times fraction numerator 1 over denominator open parentheses square root of 1 plus a squared x squared end root close parentheses end fraction equals fraction numerator a open parentheses square root of 1 plus a squared x squared end root minus 1 close parentheses over denominator 2 plus 2 a squared x squared minus 2 square root of 1 plus a squared x squared end root end fraction cross times fraction numerator 1 over denominator open parentheses square root of 1 plus a squared x squared end root close parentheses end fraction equals fraction numerator a open parentheses square root of 1 plus a squared x squared end root minus 1 close parentheses over denominator 2 square root of 1 plus a squared x squared end root open parentheses square root of 1 plus a squared x squared end root minus 1 close parentheses end fraction cross times fraction numerator 1 over denominator open parentheses square root of 1 plus a squared x squared end root close parentheses end fraction equals fraction numerator a over denominator 2 open parentheses 1 plus a squared x squared close parentheses end fraction

Answered by Dharma Teja | 25th Aug, 2014, 10:51: AM

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