Differentiate : w.r.t x. from Mathematics Continuity and Diffe

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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

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.

Let space straight y equals 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 equals 5 to the power of square root of straight x squared plus 1 end root end exponent plus left parenthesis straight x squared plus 1 right parenthesis to the power of begin inline style 5 over 2 end style end exponent therefore space dy over dx equals 5 to the power of square root of straight x squared plus 1 end root end exponent. log space 5. straight d over dx left parenthesis straight x squared plus 1 right parenthesis to the power of begin inline style 1 half end style end exponent plus 5 over 2 left parenthesis straight x squared plus 1 right parenthesis to the power of begin inline style 3 over 2 end style end exponent. straight d over dx left parenthesis straight x squared plus 1 right parenthesis space space space space space space space space space space space space space equals 5 to the power of square root of straight x squared plus 1 end root end exponent. log space 5.1 half left parenthesis straight x squared plus 1 right parenthesis to the power of begin inline style negative 1 half end style end exponent.2 straight x plus 5 over 2 left parenthesis straight x squared plus 1 right parenthesis to the power of begin inline style 3 over 2 end style end exponent.2 straight x space space space space space space space space space space space space space equals fraction numerator 5 to the power of square root of straight x squared plus 1 end root end exponent. log space 5 over denominator square root of straight x squared plus 1 end root end fraction plus 5 straight x left parenthesis straight x squared plus 1 right parenthesis to the power of begin inline style 3 over 2 end style end exponent.

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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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