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

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 Multiple Choice QuestionsShort Answer Type

231. If space straight y equals square root of straight x plus square root of straight x plus square root of straight x plus......... infinity end root end root end root comma show space that space left parenthesis 2 straight y minus 1 right parenthesis dy over dx equals 1
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232. If space straight y equals square root of log space straight x plus square root of log space straight x plus square root of log space straight x plus........ infinity end root end root end root comma space prove space that space left parenthesis 2 straight y minus 1 right parenthesis dy over dx equals 1 over straight x
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233. If space straight y square root of straight x squared plus 1 end root equals log open parentheses straight x plus square root of straight x squared plus 1 end root close parentheses comma space show space that space left parenthesis straight x squared plus 1 right parenthesis dy over dx plus xy minus 1 equals 0
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234. If space straight e to the power of straight x plus straight e to the power of straight y equals straight e to the power of straight x plus straight y comma end exponent prove space that space dy over dx equals negative fraction numerator straight e to the power of straight x left parenthesis straight e to the power of straight y minus 1 right parenthesis over denominator straight e to the power of straight y left parenthesis straight e to the power of straight y minus 1 right parenthesis end fraction
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235. If space straight y equals straight x plus fraction numerator 1 over denominator straight x plus begin display style fraction numerator 1 over denominator straight x plus... end fraction end style end fraction comma space prove space that space dy over dx equals fraction numerator straight y over denominator 2 straight y minus straight x end fraction.
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236. If space straight x square root of 1 plus straight y end root plus straight y square root of 1 plus straight x end root equals 0 comma space show space that space dy over dx equals negative left parenthesis 1 plus straight x right parenthesis to the power of negative 2 end exponent
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237. If space space ax squared plus 2 hxy plus by squared space equals space 1 comma space verify space that space dy over dx cross times dx over dy equals 1
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238. Differentiate xx w.r.t.x.
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239.

Differentiate (ax)x w.r.t.x.

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240. Differentiate the following w.r.t.x:
 open parentheses 1 over straight x close parentheses to the power of straight x


Let space straight y equals open parentheses 1 over straight x close parentheses to the power of straight x
therefore space log space straight y equals log open parentheses 1 over straight x close parentheses to the power of straight x space space space space space space space space space space space rightwards double arrow space log space straight y equals straight x space log space 1 over straight x
rightwards double arrow space log space straight y equals straight x left parenthesis log space 1 minus log space straight x right parenthesis space rightwards double arrow log space straight y equals negative left square bracket straight x space log space straight x right square bracket space space space space left square bracket because space log space 1 equals 0 right square bracket
Differentiating space both space sides space straight w. straight r. straight t space space straight x.
1 over straight y dy over dx equals negative open square brackets straight x.1 over straight x plus log space straight x.1 close square brackets
rightwards double arrow space dy over dx equals negative straight y left square bracket 1 plus log space straight x right square bracket equals negative open parentheses 1 over straight x close parentheses to the power of straight x left square bracket 1 plus log space straight x right square bracket.
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