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Continuity and Differentiability

Short 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. $If space straight y square root of straight x squared plus 1 end root equals log open parentheses straight x plus square root of straight x squared plus 1 end root close parentheses comma space show space that space left parenthesis straight x squared plus 1 right parenthesis dy over dx plus xy minus 1 equals 0$

Here 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 Differentiating space both space sides space straight w. straight r. straight t. straight x comma space we space get comma straight y. fraction numerator 2 straight x over denominator 2 square root of straight x squared plus 1 end root end fraction plus square root of straight x squared plus 1 end root dy over dx equals fraction numerator 1 over denominator straight x plus square root of straight x squared plus 1 end root end fraction cross times open square brackets 1 plus fraction numerator 2 straight x over denominator 2 square root of straight x squared plus 1 end root end fraction close square brackets space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because straight d over dx left square bracket space left curly bracket space straight f left parenthesis straight x right parenthesis space right curly bracket to the power of straight n space right square bracket equals straight n left curly bracket space left square bracket space straight f left parenthesis straight x right parenthesis space right square bracket space right curly bracket to the power of straight n minus 1 end exponent space straight f apostrophe left parenthesis straight x right parenthesis close square brackets therefore fraction numerator xy over denominator square root of straight x squared plus 1 end root end fraction plus square root of straight x squared plus 1 end root dy over dx equals fraction numerator 1 over denominator straight x plus square root of straight x squared plus 1 end root end fraction cross times fraction numerator square root of straight x squared plus 1 end root plus 1 over denominator square root of straight x squared plus 1 end root end fraction therefore fraction numerator xy over denominator square root of straight x squared plus 1 end root end fraction plus square root of straight x squared plus 1 end root dy over dx equals fraction numerator 1 over denominator square root of straight x squared plus 1 end root end fraction comma Multiplying space both space sides space by space square root of straight x squared plus 1 end root comma space space space space xy plus left parenthesis straight x squared plus 1 right parenthesis dy over dx equals 1 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space or 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 x^x w.r.t.x.

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239.

Differentiate (a^x)^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

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