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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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Here 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 therefore space space space space space space straight y equals straight x plus 1 over straight y therefore space space space space straight y squared equals straight x space straight y plus 1 Differentiating space both space sides space straight w. straight r. straight t. space straight x comma space we space get comma 2 straight y dy over dx equals straight x dy over dx plus straight y.1 plus 0 therefore space left parenthesis 2 straight y minus straight x right parenthesis dy over dx equals straight y therefore 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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