from Mathematics Continuity and Differentiability

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Continuity and Differentiability

Long Answer Type

191.

If space straight y equals square root of straight x squared plus 1 end root minus log open parentheses 1 over straight x plus square root of 1 plus 1 over straight x squared end root close parentheses comma space find space dy over dx.

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Short Answer Type

192.

If space straight y equals fraction numerator straight e to the power of straight x minus straight e to the power of negative straight x end exponent over denominator straight e to the power of straight x plus straight e to the power of negative straight x end exponent end fraction comma space prove space that space dy over dx equals 1 minus straight y squared.

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Long Answer Type

193.

If space straight y equals fraction numerator 3 minus straight u over denominator 2 plus straight u end fraction comma straight u equals fraction numerator 4 straight x over denominator 1 minus straight x squared end fraction space find space dy over dx

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Short Answer Type

194.

Find space dy over dx space when space straight y equals straight v to the power of 4 over 4 space and space straight v equals 2 over 3 straight x cubed plus 7

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195. Using Mathematical Induction, prove that

straight d over dx left parenthesis straight x to the power of straight n right parenthesis equals nx to the power of straight n minus 1 end exponent comma space for all space straight n element of straight N.

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196.

If space straight y equals open vertical bar table row cell straight f left parenthesis straight x right parenthesis end cell cell straight g left parenthesis straight x right parenthesis end cell cell straight h left parenthesis straight x right parenthesis end cell row straight l straight m straight n row straight a straight b straight c end table close vertical bar comma space prove space that space dy over dx equals open vertical bar table row cell straight f apostrophe left parenthesis straight x right parenthesis end cell cell straight g apostrophe left parenthesis straight x right parenthesis end cell cell straight h apostrophe left parenthesis straight x right parenthesis end cell row straight l straight m straight n row straight a straight b straight c end table close vertical bar

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197. $Find space dy over dx space when space straight x equals straight a space straight t squared comma space straight y equals 2 at.$

Here space straight x equals at squared space space space space space rightwards double arrow dx over dt equals 2 at Also space straight y equals 2 at space space space space space rightwards double arrow dy over dt equals 2 straight a Now space dy over dt equals fraction numerator dy over dt over denominator dx over dt end fraction fraction numerator 2 straight a over denominator 2 at end fraction equals 1 over straight t

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198.

Find space dy over dx space when space straight x equals ct comma space straight y equals straight c over straight t

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199.

If space straight y equals straight t squared plus straight t cubed space and space straight x equals straight t minus straight t to the power of 4 comma space find space dy over dx

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200.

Find space dy over dx space where space straight x equals 2 at squared comma space straight y equals at to the power of 4

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