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Continuity and Differentiability

Short Answer Type

251.

If space straight x to the power of straight y equals straight y to the power of straight x comma space prove space that space dy over dx equals fraction numerator begin display style straight y over straight x end style minus log space straight y over denominator begin display style straight x over straight y end style minus log space straight x end fraction

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252.

If space straight y equals straight x to the power of straight y comma space prove space that space dy over dx equals fraction numerator straight y squared over denominator straight x left parenthesis 1 minus straight y space log space straight x right parenthesis end fraction.

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253.

Find space dy over dx when space xy equals straight e to the power of straight x minus straight y end exponent.

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254.

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255.

If space straight y equals square root of fraction numerator 1 minus straight x over denominator 1 plus straight x end fraction end root comma space prove space that space left parenthesis 1 minus straight x squared right parenthesis dy over dx plus straight y equals 0

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256.

If space straight f left parenthesis straight x right parenthesis equals open parentheses fraction numerator 3 plus straight x over denominator 1 plus straight x end fraction close parentheses to the power of 2 plus 3 straight x end exponent comma space find space straight f apostrophe left parenthesis 0 right parenthesis

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257.

Differentiate space square root of fraction numerator left parenthesis straight x minus 3 right parenthesis left parenthesis straight x squared plus 4 right parenthesis over denominator 3 straight x squared plus 4 straight x plus 5 end fraction end root straight w. straight r. straight t. straight x

79 Views

258.

Differentiate space square root of fraction numerator left parenthesis straight x minus 1 right parenthesis left parenthesis straight x minus 2 right parenthesis over denominator left parenthesis straight x minus 3 right parenthesis left parenthesis straight x minus 4 right parenthesis left parenthesis straight x minus 5 right parenthesis end fraction end root straight w. straight r. straight t. straight x

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259. Differentiate : (x + 3)² (x + 4)³ (x + 5)^4.w.r.t.x.

Let space space space space space space space space space space straight y equals left parenthesis straight x plus 3 right parenthesis to the power of 2 space end exponent left parenthesis straight x plus 4 right parenthesis to the power of 3 space end exponent left parenthesis straight x plus 5 right parenthesis to the power of 4 therefore space space space space space log space straight y equals log left square bracket left parenthesis straight x plus 3 right parenthesis to the power of 2 space end exponent left parenthesis straight x plus 4 right parenthesis to the power of 3 space end exponent left parenthesis straight x plus 5 right parenthesis to the power of 4 right square bracket therefore space space space space space space space space space space space space space space equals log left parenthesis straight x plus 3 right parenthesis to the power of 2 space end exponent log left parenthesis straight x plus 4 right parenthesis to the power of 3 space end exponent log left parenthesis straight x plus 5 right parenthesis to the power of 4 therefore space space space space space log space straight y equals 2 log left parenthesis straight x plus 3 right parenthesis plus 3 log left parenthesis straight x plus 4 right parenthesis plus 4 to the power of space log left parenthesis straight x plus 5 right parenthesis therefore 1 over straight y dy over dx equals fraction numerator 2 over denominator straight x plus 3 end fraction plus fraction numerator 3 over denominator straight x plus 4 end fraction plus fraction numerator 4 over denominator straight x plus 5 end fraction therefore space space space space space space dy over dx equals left parenthesis straight x plus 3 right parenthesis to the power of 2 space end exponent left parenthesis straight x plus 4 right parenthesis to the power of 3 space end exponent left parenthesis straight x plus 5 right parenthesis to the power of 4 open square brackets fraction numerator 2 over denominator straight x plus 3 end fraction plus fraction numerator 3 over denominator straight x plus 4 end fraction plus fraction numerator 4 over denominator straight x plus 5 end fraction close square brackets

81 Views

260.

Find the derivative of the function given by

f(x)=(1 + x)(1 + x²)(1 + x⁴)(1 + x⁸)and hence find f'(1).

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Previous Year Papers

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Test Yourself

Short Answer Type

259. Differentiate : (x + 3)2 (x + 4)3 (x + 5)4.w.r.t.x.

259. Differentiate : (x + 3)² (x + 4)³ (x + 5)^4.w.r.t.x.