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

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Continuity and Differentiability

Short Answer Type

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181. Differentiate the following w.r.t.x: $square root of log space left parenthesis straight x cubed plus straight e to the power of straight x right parenthesis end root$

Let space straight y equals square root of log space left parenthesis straight x cubed plus straight e to the power of straight x right parenthesis end root equals left square bracket log space left parenthesis straight x cubed plus straight e to the power of straight x right parenthesis right square bracket to the power of begin inline style 1 half end style end exponent therefore space dy over dx equals 1 half left square bracket log space left parenthesis straight x cubed plus straight e to the power of straight x right parenthesis right square bracket to the power of negative 1 half end exponent straight d over dx left square bracket log space left parenthesis straight x cubed plus straight e to the power of straight x right parenthesis right square bracket space space space space space space space space space space space space space equals fraction numerator 1 over denominator 2 square root of log space left parenthesis straight x cubed plus straight e to the power of straight x right parenthesis end root end fraction cross times fraction numerator 1 over denominator straight x cubed plus straight e to the power of straight x end fraction cross times straight d over dx left parenthesis straight x cubed plus straight e to the power of straight x right parenthesis space space space space space space space space space space space space space equals fraction numerator 1 over denominator 2 square root of log space left parenthesis straight x cubed plus straight e to the power of straight x right parenthesis end root end fraction cross times fraction numerator 1 over denominator straight x cubed plus straight e to the power of straight x end fraction cross times left parenthesis 3 straight x squared plus straight e to the power of straight x right parenthesis equals fraction numerator 3 straight x squared plus straight e to the power of straight x over denominator 2 left parenthesis straight x cubed plus straight e to the power of straight x right parenthesis square root of log space left parenthesis straight x cubed plus straight e to the power of straight x right parenthesis end root end fraction

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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

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

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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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