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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.

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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

Let space straight y equals log left square bracket left parenthesis straight x plus 2 right parenthesis left parenthesis straight x cubed minus straight x right parenthesis right square bracket therefore space space straight y equals log left parenthesis straight x plus 2 right parenthesis plus log left parenthesis straight x cubed minus straight x right parenthesis therefore space dy over dx equals fraction numerator 1 over denominator straight x plus 2 end fraction. straight d over dx left parenthesis straight x plus 2 right parenthesis plus fraction numerator 1 over denominator straight x cubed minus straight x end fraction. straight d over dx left parenthesis straight x cubed minus straight x right parenthesis space space space space space space space space space space space space space equals fraction numerator 1 over denominator straight x plus 2 end fraction cross times 1 plus fraction numerator 1 over denominator straight x cubed minus straight x end fraction cross times left parenthesis 3 straight x squared minus 1 right parenthesis equals fraction numerator 1 over denominator straight x plus 2 end fraction plus fraction numerator 3 straight x squared minus 1 over denominator straight x cubed minus straight x end fraction

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177.

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