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Continuity and Differentiability

Short Answer Type

171.

Differentiate $straight e to the power of log open parentheses straight x plus square root of straight x squared plus straight a squared end root close parentheses end exponent$ w.r.t x.

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Long Answer Type

172.
Differentiate $fraction numerator square root of straight a plus straight x end root minus square root of straight a minus straight x end root over denominator square root of bold a bold plus bold x end root plus square root of bold a bold minus bold x end root end fraction$ w.r.t x.

Let space straight y equals fraction numerator square root of straight a plus straight x end root minus square root of straight a minus straight x end root over denominator square root of bold a bold plus bold x end root plus square root of bold a bold minus bold x end root end fraction equals fraction numerator square root of straight a plus straight x end root minus square root of straight a minus straight x end root over denominator square root of bold a bold plus bold x end root plus square root of bold a bold minus bold x end root end fraction cross times fraction numerator square root of straight a plus straight x end root minus square root of straight a minus straight x end root over denominator square root of bold a bold plus bold x end root minus square root of bold a bold minus bold x end root end fraction equals fraction numerator left parenthesis straight a plus straight x right parenthesis plus left parenthesis straight a minus straight x right parenthesis minus 2 square root of straight a plus straight x end root square root of straight a minus straight x end root over denominator left parenthesis straight a plus straight x right parenthesis minus left parenthesis straight a minus straight x right parenthesis end fraction equals fraction numerator 2 straight a minus 2 square root of straight a squared minus straight x squared end root over denominator 2 straight x end fraction equals fraction numerator straight a minus square root of straight a squared minus straight x squared end root over denominator straight x end fraction therefore space dy over dx equals fraction numerator straight x begin display style straight d over dx end style open parentheses straight a minus square root of straight a squared minus straight x squared end root close parentheses minus open parentheses straight a minus square root of straight a squared minus straight x squared end root close parentheses begin display style straight d over dx end style left parenthesis straight x right parenthesis over denominator straight x squared end fraction equals fraction numerator straight x open parentheses 0 minus begin display style fraction numerator negative 2 straight x over denominator 2 square root of straight a squared minus straight x squared end root end fraction end style close parentheses minus open parentheses straight a minus square root of straight a squared minus straight x squared end root close parentheses.1 over denominator straight x squared end fraction equals fraction numerator begin display style fraction numerator straight x squared over denominator square root of straight a squared minus straight x squared end root end fraction end style minus straight a plus square root of straight a squared minus straight x squared end root over denominator straight x squared end fraction equals fraction numerator straight x squared minus straight a square root of straight a squared minus straight x squared end root plus straight a squared minus straight x squared over denominator straight x squared square root of straight a squared minus straight x squared end root end fraction equals fraction numerator straight a squared minus straight a square root of straight a squared minus straight x squared end root over denominator straight x squared square root of straight a squared minus straight x squared end root end fraction

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173.

Differentiate $square root of a italic plus square root of a italic plus square root of a italic plus x to the power of italic 2 end root end root end root$ w.r.t x

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Short Answer Type

174.

If space straight y equals square root of straight a plus square root of straight a plus straight x end root end root comma space find space dy over dx.

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175.

If space straight y equals square root of 4 plus square root of 4 plus square root of 4 plus straight x squared end root end root end root comma space find space dy over dx.

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176. Differentiate log [(x + 2)(x³ - x)] w.r.t.x

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177.

Differentiate $log open parentheses fraction numerator straight x squared plus straight x plus 1 over denominator straight x squared minus straight x plus 1 end fraction close parentheses$