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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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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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We space have space ax squared plus 2 hxy plus by squared equals 1... left parenthesis 1 right parenthesis Differentiating space both space sides space of space left parenthesis 1 right parenthesis comma space straight w. straight r. straight t. straight x space regarding space straight y space as space straight a space function space of space straight x. space we space get comma 2 ax plus 2 straight h open parentheses straight x. dy over dx plus straight y.1 close parentheses plus straight b.2 straight y dy over dx equals 0 or space ax plus hx dy over dx plus hy plus by dy over dx equals 0 or space dy over dx equals negative fraction numerator ax plus hy over denominator hx plus by end fraction 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 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.. left parenthesis 2 right parenthesis space

Again Differentiating both sides of (1), w.r.t. y regarding x as a function of y, we get,

space space space space space 2 straight a. straight x dx over dy plus 2 straight h open parentheses straight x.1 plus straight y dx over dy close parentheses plus 2 by equals 0 or space ax plus dx over dy plus hx plus hy dx over dy plus by equals 0 therefore space dx over dy left square bracket ax plus hy right square bracket equals negative left parenthesis hx plus by right parenthesis space or space dy over dx equals negative fraction numerator hx plus bx over denominator ax plus hy end fraction 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... left parenthesis 3 right parenthesis Multiplying space left parenthesis 2 right parenthesis space and space left parenthesis 3 right parenthesis comma space we space get comma dy over dx cross times dx over dy equals negative fraction numerator ax plus hy over denominator hx plus hy end fraction cross times negative fraction numerator hx plus bx over denominator ax plus hy end fraction 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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