Differentiate the following w.r.t.x: from Mathematics Continu

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Continuity and Differentiability

Short Answer Type

181. Differentiate the following w.r.t.x:

square root of log space left parenthesis straight x cubed plus straight e to the power of straight x right parenthesis end root

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182. Differentiate the following w.r.t.x:

square root of log space left parenthesis logx right parenthesis end root

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183. Differentiate the following w.r.t.x:

log open square brackets straight e to the power of 3 straight x end exponent open parentheses fraction numerator 5 straight x minus 3 over denominator 4 straight x plus 2 end fraction close parentheses to the power of 1 third end exponent close square brackets

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184. Differentiate the following w.r.t.x:

log open parentheses straight x plus 2 plus square root of straight x squared plus 4 straight x plus 1 end root close parentheses

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185. Differentiate the following w.r.t.x:

log open parentheses straight x plus square root of 1 plus straight x squared end root close parentheses

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186. Differentiate the following w.r.t.x:

log open parentheses square root of straight x minus straight a end root plus square root of straight x minus straight b end root close parentheses

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Long Answer Type

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187. Differentiate the following w.r.t.x: $log open parentheses fraction numerator square root of straight a plus straight x end root plus square root of straight a minus straight x end root over denominator square root of straight a plus straight x end root minus square root of straight a minus straight x end root end fraction close parentheses$

Let space straight y equals log open parentheses fraction numerator square root of straight a plus straight x end root plus square root of straight a minus straight x end root over denominator square root of straight a plus straight x end root minus square root of straight a minus straight x end root end fraction close parentheses space space space space space space equals log open square brackets fraction numerator square root of straight a plus straight x end root plus square root of straight a minus straight x end root over denominator square root of straight a plus straight x end root minus square root of straight a minus straight x end root end fraction cross times fraction numerator square root of straight a plus straight x end root plus square root of straight a minus straight x end root over denominator square root of straight a plus straight x end root plus square root of straight a minus straight x end root end fraction close square brackets space space space space space space equals log open square brackets fraction numerator straight a plus straight x plus straight a minus straight x plus 2 square root of straight a squared minus straight x squared end root over denominator straight a plus straight x minus straight x left parenthesis straight a minus straight x right parenthesis end fraction close square brackets equals log open parentheses fraction numerator straight a plus square root of straight a squared minus straight x squared end root over denominator straight x end fraction close parentheses space space space space space space equals log open parentheses straight a plus square root of straight a squared minus straight x squared end root close parentheses minus log space straight x therefore dy over dx equals fraction numerator 1 over denominator straight a plus square root of straight a squared minus straight x squared end root end fraction cross times straight d over dx left square bracket straight a plus left parenthesis straight a squared minus straight x squared right parenthesis right square bracket to the power of begin inline style 1 half end style end exponent minus 1 over straight x space space space space space space equals fraction numerator 1 over denominator straight a plus square root of straight a squared minus straight x squared end root end fraction cross times 1 half left parenthesis straight a squared minus straight x squared right parenthesis to the power of begin inline style 1 half end style minus 1 end exponent cross times straight d over dx left parenthesis straight a squared minus straight x squared right parenthesis minus 1 over straight x space space space space space space equals fraction numerator 1 over denominator straight a plus square root of straight a squared minus straight x squared end root end fraction cross times fraction numerator 1 over denominator 2 square root of straight a squared minus straight x squared end root end fraction left parenthesis negative 2 straight x right parenthesis minus 1 over straight x space space space space space space equals fraction numerator negative straight x over denominator square root of straight a squared minus straight x squared end root. open square brackets straight a plus square root of straight a squared minus straight x squared end root close square brackets end fraction minus 1 over straight x equals negative fraction numerator straight x squared plus 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 square root of straight a squared minus straight x squared end root open square brackets straight a plus square root of straight a squared minus straight x squared end root close square brackets end fraction space space space space space space equals fraction numerator straight a open square brackets straight a plus square root of straight a squared minus straight x squared end root close square brackets over denominator straight x square root of straight a squared minus straight x squared end root open square brackets straight a plus square root of straight a squared minus straight x squared end root close square brackets end fraction equals negative fraction numerator straight a over denominator straight x square root of straight a squared minus straight x squared end root end fraction

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Short Answer Type

188. Differentiate the following w.r.t.x:

open parentheses fraction numerator straight x plus square root of straight x squared minus straight a squared end root over denominator straight x minus square root of straight x squared minus straight a squared end root end fraction close parentheses

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189. Differentiate :

5 to the power of square root of straight x squared plus 1 end root end exponent plus open parentheses square root of straight x squared plus 1 end root close parentheses to the power of 5

w.r.t x.

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190. Differentiate :

straight x square root of straight x squared plus 1 end root plus log open parentheses square root of straight x squared plus 1 end root close parentheses

w.r.t x.

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