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Continuity and Differentiability

Short Answer Type

201.

Find space dy over dx space where space straight x equals 4 straight t comma space straight y equals 4 over straight t

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202.

If space straight x equals straight a open parentheses fraction numerator 1 plus straight t squared over denominator 1 minus straight t squared end fraction close parentheses space and space straight y equals fraction numerator 2 straight t over denominator 1 minus straight t squared end fraction comma space find space dy over dx.

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203.

If space straight x equals fraction numerator 1 minus straight t squared over denominator 1 plus straight t squared end fraction comma space straight y equals fraction numerator 2 straight t over denominator 1 plus straight t squared end fraction comma space prove space that space dy over dx plus straight x over straight y equals 0.

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204.

Find space dy over dx space where space straight x equals fraction numerator straight a left parenthesis 1 minus straight t squared right parenthesis over denominator 1 minus straight t squared end fraction comma space straight y space equals fraction numerator 2 bt over denominator 1 minus straight t squared end fraction

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205.

Find space dy over dx space when space straight x equals straight e to the power of straight t. space log space straight t comma space straight y equals straight t. log space straight t.

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206.

Find space dy over dx when space straight x equals fraction numerator 3 at over denominator 1 plus straight t cubed end fraction comma space straight y equals fraction numerator 3 at squared over denominator 1 plus straight t cubed end fraction

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207. $If space space straight x equals fraction numerator 3 at over denominator 1 plus straight t squared end fraction comma space straight y equals fraction numerator 3 at squared over denominator 1 plus straight t squared end fraction comma space then space find space dy over dt space at space straight t equals 2$

straight x equals fraction numerator 3 at over denominator 1 plus straight t squared end fraction comma space straight y equals fraction numerator 3 at squared over denominator 1 plus straight t squared end fraction therefore space dx over dt equals 3 straight a straight d over dt open square brackets fraction numerator straight t over denominator 1 plus straight t squared end fraction close square brackets equals 3 straight a open square brackets fraction numerator left parenthesis 1 plus straight t squared right parenthesis.1 minus straight t. space 2 straight t over denominator left parenthesis 1 plus straight t squared right parenthesis squared end fraction close square brackets equals 3 straight a open square brackets fraction numerator 1 plus straight t squared minus 2 straight t squared over denominator left parenthesis 1 plus straight t squared right parenthesis squared end fraction close square brackets therefore space dx over dt equals fraction numerator 3 straight a left parenthesis 1 plus straight t squared right parenthesis over denominator left parenthesis 1 plus straight t squared right parenthesis squared end fraction Also space dy over dt equals 3 straight a straight d over dt open square brackets fraction numerator straight t squared over denominator 1 plus straight t squared end fraction close square brackets equals 3 straight a open square brackets fraction numerator left parenthesis 1 plus straight t squared right parenthesis.2 straight t minus straight t squared.2 straight t over denominator left parenthesis 1 plus straight t squared right parenthesis squared end fraction close square brackets space space space space space space space space space space space space space space space space space equals 3 straight a open square brackets fraction numerator 2 straight t plus 2 straight t cubed minus 2 straight t cubed over denominator left parenthesis 1 plus straight t squared right parenthesis squared end fraction close square brackets equals fraction numerator 3 straight a left parenthesis 2 straight t right parenthesis over denominator left parenthesis 1 plus straight t squared right parenthesis squared end fraction therefore space dy over dt equals fraction numerator 6 at over denominator left parenthesis 1 plus straight t squared right parenthesis squared end fraction Now space dy over dt equals fraction numerator begin display style dy over dt end style over denominator begin display style dx over dt end style end fraction equals fraction numerator 6 at over denominator left parenthesis 1 plus straight t squared right parenthesis squared end fraction cross times fraction numerator left parenthesis 1 plus straight t squared right parenthesis squared over denominator 3 straight a left parenthesis 1 plus straight t squared right parenthesis end fraction therefore space dy over dt equals fraction numerator 2 straight t over denominator 1 minus straight t squared end fraction At space straight t equals 2 comma space dy over dx equals fraction numerator 2 cross times 2 over denominator 1 minus 4 end fraction equals fraction numerator 4 over denominator negative 3 end fraction equals negative 4 over 3.

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Long Answer Type

208.

If space straight x equals fraction numerator 1 plus log space straight t over denominator straight t squared end fraction comma space straight y equals fraction numerator 3 plus 2 space log space straight t over denominator straight t end fraction comma space straight t greater than 0 comma space prove space that space straight y dy over dx minus 2 straight x open parentheses dy over dx close parentheses squared equals 1

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Short Answer Type

209.

For space straight a space positive space constant space straight a space find space dy over dx comma space where space straight y equals straight a to the power of straight t plus 1 over straight t end exponent comma space and space straight x equals open parentheses straight t plus 1 over straight t close parentheses to the power of straight a.

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210. Differentiate 7x⁵ - 11 x² w.r.t. 7x² - 15 x

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