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Continuity and Differentiability

Short Answer Type

511.

Find space dy over dx space when space straight y equals left parenthesis sin space straight x right parenthesis to the power of straight x plus sin to the power of negative 1 end exponent square root of straight x.

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512. $If space straight y equals cos to the power of negative 1 end exponent open parentheses fraction numerator 3 plus 5 space cos space straight x over denominator 5 plus 3 space cos space straight x end fraction close parentheses comma space prove space that space cos space straight x equals fraction numerator 4 minus 5 straight y subscript 1 over denominator 3 straight y subscript 1 end fraction comma space where space straight y subscript 1 equals dy over dx$

space space space space space space space space space space space space straight y equals cos to the power of negative 1 end exponent open parentheses fraction numerator 3 plus 5 space cos space straight x over denominator 5 plus 3 space cos space straight x end fraction close parentheses space space therefore space dy over dx equals negative fraction numerator 1 over denominator square root of 1 minus open parentheses begin display style fraction numerator 3 plus 5 space cos space straight x over denominator 5 plus 3 space cos space straight x end fraction end style close parentheses squared end root end fraction. straight d over dx open parentheses fraction numerator 3 plus 5 space cos space straight x over denominator 5 plus 3 space cos space straight x end fraction close parentheses equals negative fraction numerator 5 plus 3 cos space straight x over denominator square root of left parenthesis 5 plus 3 cos space straight x right parenthesis squared minus left parenthesis 3 plus 5 space cos space straight x right parenthesis squared end root end fraction cross times fraction numerator left parenthesis 5 plus 3 cos space straight x right parenthesis left parenthesis negative 5 sin space straight x right parenthesis minus left parenthesis 3 plus 5 space cos space straight x right parenthesis left parenthesis negative 3 sin space straight x right parenthesis over denominator left parenthesis 5 plus 3 cos space straight x right parenthesis squared end fraction equals negative fraction numerator 1 over denominator square root of 25 plus 9 space cos squared space straight x minus 9 minus 25 space cos to the power of 2 space end exponent straight x end root end fraction cross times fraction numerator negative 25 space sin space straight x minus 15 space sin space straight x space cos space straight x plus 9 space sin space straight x plus 15 space sin space straight x space cos space straight x over denominator 5 plus 3 cos space straight x end fraction equals negative fraction numerator 1 over denominator square root of 16 minus 16 space cos squared space straight x end root end fraction cross times fraction numerator negative 16 space sin space straight x over denominator 5 plus 3 space cos space straight x end fraction equals negative fraction numerator 1 over denominator 4 space sin space straight x end fraction cross times fraction numerator negative 16 space sin space straight x over denominator 5 plus 3 space cos space straight x end fraction space space space space space therefore space straight y subscript 1 equals fraction numerator 4 over denominator 5 plus 3 space cos space straight x end fraction space space space straight R. straight H. straight S. equals fraction numerator 4 minus 5 straight y subscript 1 over denominator 3 straight y subscript 1 end fraction equals fraction numerator 4 minus begin display style fraction numerator 20 over denominator 5 plus 3 space cos space straight x end fraction end style over denominator begin display style fraction numerator 12 over denominator 5 plus 3 space cos space straight x end fraction end style end fraction equals fraction numerator 20 plus 12 space cos space straight x minus 20 over denominator 12 end fraction space space space space space space space space space space equals fraction numerator 12 space cos space straight x over denominator 12 end fraction equals cos space straight x equals straight L. straight H. straight S.

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513.

Expressing space the space equation space in space terms space of space an space equation space square root of 1 minus straight x squared end root plus square root of 1 minus straight y squared end root equals straight a left parenthesis straight x minus straight y right parenthesis space in space term space oa space an space equation involving space inverse space trigonometric space functions space by space straight a space suitable space substitution comma space prove space that dy over dx equals fraction numerator square root of 1 minus straight y squared end root over denominator square root of 1 minus straight x squared end root end fraction.

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514. Find the derivative w.r.t.x of the function

left parenthesis log subscript cos space straight x end subscript sin space straight x right parenthesis space left parenthesis log subscript sin space straight x end subscript space cos space straight x right parenthesis to the power of negative 1 end exponent plus sin to the power of negative 1 end exponent open parentheses fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction close parentheses space at space straight x equals straight pi over 4

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515.

Differentiate space open parentheses log subscript cos space straight x end subscript space sin space straight x close parentheses open parentheses log subscript sin space straight x end subscript space cos space straight x close parentheses plus cos to the power of negative 1 end exponent open parentheses fraction numerator 1 minus straight x squared over denominator 1 plus straight x squared end fraction close parentheses space straight w. straight r. straight t. straight x space at space straight x equals straight pi over 4.

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516.

Prove space that space straight d over dx open curly brackets fraction numerator 1 over denominator 4 square root of 2 end fraction log fraction numerator straight x squared plus straight x square root of 2 plus 1 over denominator straight x squared minus straight x square root of 2 plus 1 end fraction plus fraction numerator 1 over denominator 2 square root of 2 end fraction tan to the power of negative 1 end exponent fraction numerator straight x square root of 2 over denominator 1 minus straight x squared end fraction close curly brackets equals fraction numerator 1 over denominator 1 plus straight x to the power of 4 end fraction

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517. Find the second order derivatives of the following functions : x² + 3x + 2

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518. Find the second order derivatives of the following functions : x²⁰

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519. Find the second order derivatives of the following functions :x cos x

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520. Find the second order derivatives of the following functions :log x

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