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

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

Here space straight y equals fraction numerator 3 minus straight u over denominator 2 plus straight u end fraction therefore space dy over dx equals fraction numerator left parenthesis 2 plus straight u right parenthesis begin display style straight d over du end style left parenthesis 3 minus straight u right parenthesis minus left parenthesis 3 minus straight u right parenthesis begin display style straight d over du end style left parenthesis 2 plus straight u right parenthesis over denominator left parenthesis 2 plus straight u right parenthesis squared end fraction equals fraction numerator left parenthesis 2 plus straight u right parenthesis left parenthesis negative 1 right parenthesis minus left parenthesis 3 minus straight u right parenthesis.1 over denominator left parenthesis 2 plus straight u right parenthesis squared end fraction space space space space space space space space space space space space equals fraction numerator 2 minus straight u minus 3 plus straight u over denominator left parenthesis 2 plus straight u right parenthesis squared end fraction therefore space dy over dx equals negative fraction numerator 5 over denominator left parenthesis 2 plus straight u right parenthesis squared 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 space space space... left parenthesis 1 right parenthesis Also space straight u equals fraction numerator 4 straight x over denominator 1 minus straight x squared end fraction equals 4 open square brackets fraction numerator straight x over denominator 1 minus straight x squared end fraction close square brackets therefore space du over dx equals 4 open curly brackets fraction numerator left parenthesis 1 minus straight x squared right parenthesis begin display style straight d over dx end style left parenthesis straight x right parenthesis minus straight x. begin display style straight d over dx end style left parenthesis 1 minus straight x squared right parenthesis over denominator left parenthesis 1 minus straight x squared right parenthesis squared end fraction close curly brackets space space space space space space space space space space space space space equals 4 open square brackets fraction numerator left parenthesis 1 minus straight x squared right parenthesis.1 minus straight x. left parenthesis negative 2 straight x right parenthesis over denominator left parenthesis 1 minus straight x squared right parenthesis squared end fraction close square brackets equals 4 open square brackets fraction numerator 1 minus straight x squared plus 2 straight x squared over denominator left parenthesis 1 minus straight x squared right parenthesis squared end fraction close square brackets therefore space du over dx equals fraction numerator 4 left parenthesis 1 minus straight x squared right parenthesis over denominator left parenthesis 1 minus straight x squared right parenthesis squared 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 space space space space space space... left parenthesis 2 right parenthesis Now space dy over dx equals dy over du. du over dx equals negative fraction numerator 5 over denominator left parenthesis 2 plus straight u right parenthesis squared end fraction. fraction numerator 4 left parenthesis 1 plus straight x squared right parenthesis over denominator left parenthesis 1 minus straight x squared right parenthesis squared end fraction space space space space space space space space space space space space space space space space space space space space space space left square bracket because space of space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis right square bracket space space space space space space space space space space space space space space space space space equals fraction numerator negative 5 left parenthesis 1 minus straight x squared right parenthesis squared over denominator 4 left parenthesis 1 plus 2 straight x minus straight x squared right parenthesis squared end fraction. fraction numerator 4 left parenthesis 1 plus straight x squared right parenthesis over denominator left parenthesis 1 minus straight x squared right parenthesis squared end fraction equals negative fraction numerator 5 left parenthesis 1 minus straight x squared right parenthesis squared over denominator left parenthesis 1 plus 2 straight x minus straight x squared right parenthesis squared end fraction

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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.

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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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