from Mathematics Continuity and Differentiability

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Continuity and Differentiability

Short Answer Type

421.

Differentiate space straight w. straight r. straight t. straight x colon space straight y equals sin to the power of negative 1 end exponent open parentheses fraction numerator 1 plus 2 sin space straight x over denominator 2 plus cos space straight x end fraction close parentheses

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422. $Differentiate space sin to the power of negative 1 end exponent open parentheses fraction numerator straight a plus straight b space cos space straight x over denominator straight b plus straight a space cos space straight x end fraction close parentheses space straight w. straight r. straight t. straight x comma space using space chain space rule$

Let space space space space space space space space space straight y equals sin to the power of negative 1 end exponent open parentheses fraction numerator straight a plus straight b space cos space straight x over denominator straight b plus straight a space cos space straight x end fraction close parentheses Put space fraction numerator straight a plus straight b space cos space straight x over denominator straight b plus straight a space cos space straight x end fraction equals straight u therefore space space space space space space space space space space space straight y equals sin to the power of negative 1 end exponent therefore space space space space space dy over du equals fraction numerator 1 over denominator square root of 1 minus straight u squared end root end fraction Also space du over dx equals fraction numerator left parenthesis straight b plus straight a space cos space straight x right parenthesis left parenthesis negative straight b space sin space straight x right parenthesis minus left parenthesis straight a plus straight b space cos space straight x right parenthesis left parenthesis negative straight a space sin space straight x right parenthesis over denominator left parenthesis straight b plus straight a space cos space straight x right parenthesis squared end fraction space space space space space space space space space space space space space space space space space equals fraction numerator negative straight b squared sin space straight x minus absin space straight x space cos space straight x plus straight a squared sin space straight x space plus absin space straight x space cos space straight x over denominator left parenthesis straight b plus straight a space cos space straight x right parenthesis squared end fraction therefore space space space space space du over dx equals fraction numerator left parenthesis straight a squared minus straight b squared right parenthesis sin space straight x over denominator left parenthesis straight b plus straight a space cos space straight x right parenthesis squared end fraction Now space by space chain space rule comma

dy over dx equals dy over du. du over dx equals fraction numerator 1 over denominator square root of 1 minus straight u squared end root end fraction. fraction numerator left parenthesis straight a squared minus straight b squared right parenthesis sin space straight x over denominator left parenthesis straight b plus acos space straight x right parenthesis end fraction space left square bracket because space of left parenthesis 1 right parenthesis comma left parenthesis 2 right parenthesis right square bracket space space space space space space space space equals fraction numerator 1 over denominator square root of 1 minus begin display style fraction numerator left parenthesis straight a plus straight b space cos space straight x right parenthesis squared over denominator left parenthesis straight b plus acos space straight x right parenthesis squared end fraction end style end root end fraction. fraction numerator left parenthesis straight a squared minus straight b squared right parenthesis sin space straight x over denominator left parenthesis straight b plus acos space straight x right parenthesis squared end fraction space space space space space space space space equals fraction numerator straight b plus acos space straight x over denominator square root of left parenthesis straight b plus acos space straight x right parenthesis squared minus left parenthesis straight a plus straight b space cos space straight x right parenthesis squared end root end fraction cross times fraction numerator left parenthesis straight a squared minus straight b squared right parenthesis sin space straight x over denominator left parenthesis straight b plus acos space straight x right parenthesis squared end fraction space space space space space space space space equals fraction numerator 1 over denominator square root of left parenthesis straight b squared minus straight a squared right parenthesis minus left parenthesis straight b squared minus straight a squared right parenthesis cos squared straight x end root end fraction cross times fraction numerator left parenthesis straight a squared minus straight b squared right parenthesis sin space straight x over denominator straight b plus acos space straight x end fraction space space space space space space space space equals fraction numerator 1 over denominator square root of straight b squared minus straight a squared end root cross times sin space straight x end fraction cross times fraction numerator negative left parenthesis straight b squared minus straight a squared right parenthesis sin space straight x over denominator straight b plus acos space straight x end fraction equals negative fraction numerator square root of straight b squared minus straight a squared end root over denominator straight b plus acos space straight x end fraction

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423.

Prove space that space straight d over straight d open square brackets square root of 1 minus straight x squared end root sin to the power of negative 1 end exponent straight x minus straight x close square brackets equals negative fraction numerator straight x space sin to the power of negative 1 end exponent straight x over denominator square root of 1 minus straight x squared end root end fraction

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424.

Prove space that space straight d over dx left square bracket 2 straight x space tan to the power of negative 1 end exponent straight x minus log left parenthesis 1 plus straight x squared right parenthesis right square bracket equals 2 space tan to the power of negative 1 end exponent straight x.

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425.

Prove space that space straight d over dx open square brackets straight x over 2 square root of straight a squared minus straight x squared end root plus straight a squared over 2 sin to the power of negative 1 end exponent straight x over straight a close square brackets equals square root of straight a squared minus straight x squared end root

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426.

Differerntiate space cos to the power of negative 1 end exponent open parentheses fraction numerator 3 cos space straight x minus 4 sin space straight x over denominator 5 end fraction close parentheses space straight w. straight r. straight t. straight x.

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427.

Find space dy over dx space if space straight y equals sin to the power of negative 1 end exponent open parentheses fraction numerator 5 straight x plus 12 square root of 1 minus straight x squared end root over denominator 13 end fraction close parentheses

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428.

If space straight y equals sec to the power of negative 1 end exponent open parentheses fraction numerator straight x plus 1 over denominator straight x minus 1 end fraction close parentheses plus sin to the power of negative 1 end exponent open parentheses fraction numerator straight x plus 1 over denominator straight x minus 1 end fraction close parentheses comma space find space dy over dx.

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429. Differentiate the following w.r.t.x:

tan to the power of negative 1 end exponent open square brackets open parentheses fraction numerator 1 minus cos space straight x over denominator 1 plus cos space straight x end fraction close parentheses to the power of 1 half end exponent close square brackets

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430. Differentiate the following w.r.t.x:

tan to the power of negative 1 end exponent open parentheses square root of fraction numerator 1 plus sin space straight x over denominator 1 minus sin space straight x end fraction end root close parentheses

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