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Continuity and Differentiability

Short Answer Type

541. If y = 3e^2x + 2e^3x, then prove that

fraction numerator straight d squared straight y over denominator dx squared end fraction minus 5 dy over dx plus 6 straight y equals 0.

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542.

If space straight y space equals straight A space straight e to the power of straight m space straight x end exponent plus straight B space straight e to the power of straight n space straight x end exponent comma space show space that space dy over dx minus left parenthesis straight m plus straight n right parenthesis dy over dx plus straight m space straight n space straight y equals 0

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543.

If space straight y equals straight a space straight e to the power of mx plus straight b space straight e to the power of negative mx end exponent comma space prove space that space fraction numerator straight d squared straight y over denominator dx squared end fraction minus straight m squared space straight y equals 0

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544.

If space straight y equals 500 space straight e to the power of 7 straight x end exponent plus 600 space straight e to the power of negative 7 straight x end exponent comma space show space that space fraction numerator straight d squared straight y over denominator dx squared end fraction equals 49 space straight y.

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545. $If space straight e to the power of straight y left parenthesis straight x plus 1 right parenthesis equals 1 comma space show space that space fraction numerator straight d squared straight y over denominator dx squared end fraction equals open parentheses dy over dx close parentheses squared$

Here space space space straight e to the power of straight y left parenthesis straight x plus 1 right parenthesis equals 1 therefore space space space log left square bracket straight e to the power of straight y left parenthesis straight x plus 1 right parenthesis right square bracket equals log space 1 therefore space space space log space straight e to the power of straight y left parenthesis straight x plus 1 right parenthesis equals 0 therefore space space space space space space space space space space straight y space log space straight e equals negative log left parenthesis straight x plus 1 right parenthesis therefore space space space space space space space space space space space space space space space space space space space space straight y equals negative log left parenthesis straight x plus 1 right parenthesis 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 square bracket because space loge equals 1 right square bracket therefore space space space space space space space space space space space space space space dy over dx equals negative fraction numerator 1 over denominator straight x plus 1 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 space space... left parenthesis 1 right parenthesis therefore space space space space space space space space space space space space space fraction numerator straight d squared straight y over denominator dx squared end fraction equals negative open square brackets fraction numerator left parenthesis straight x plus 1 right parenthesis. begin display style straight d over dx end style left parenthesis 1 right parenthesis minus 1. begin display style straight d over dx end style left parenthesis straight x plus 1 right parenthesis over denominator left parenthesis straight x plus 1 right parenthesis squared end fraction close square brackets equals negative open square brackets fraction numerator left parenthesis straight x plus 1 right parenthesis.0 minus 1.1 over denominator left parenthesis straight x plus 1 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 space space space space space space space space space space equals fraction numerator 1 over denominator left parenthesis straight x plus 1 right parenthesis squared end fraction equals open parentheses negative fraction numerator 1 over denominator straight x plus 1 end fraction close parentheses squared therefore space space space space space space space space space space space space space space fraction numerator straight d squared straight y over denominator dx squared end fraction equals open parentheses dy over dx close parentheses squared 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 square bracket because space of space left parenthesis 1 right parenthesis right square bracket

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546. If y=sin ^-1x, prove that

dy over dx equals straight x over open parentheses 1 minus straight x squared close parentheses to the power of begin display style 3 over 2 end style end exponent

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547. If y=(log x)², prove that x²

fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight x dy over dx minus 2 equals 0

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548. If y=cos(log x)+sin(log x), then prove that x²y₂+x y₁+y=0 where y₁ and y₂are first and second order derivatives.

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549. If =3 cos(log x)+4 sin(log x), show that x²y₂+x y₁+y=0.

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550. If y = a cos (log x) + b sin (log x), show that

straight x squared fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight x dy over dx plus straight y equals 0.

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