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Continuity and Differentiability

Short Answer Type

411. Differentiate the following functions w.r.t.x:

space fraction numerator 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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412. Differentiate the following functions w.r.t.x:

cos to the power of negative 1 end exponent square root of fraction numerator 1 plus straight x over denominator 2 end fraction end root

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413. Differentiate the following functions w.r.t.x:

sin to the power of negative 1 end exponent square root of fraction numerator 1 plus straight x squared over denominator 2 end fraction end root

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414. Differentiate the following functions w.r.t.x:

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

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415. Differentiate the following functions w.r.t.x

fraction numerator cos left parenthesis log space straight x right parenthesis plus tan to the power of negative 1 end exponent straight x over denominator straight e to the power of straight x end fraction

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416. Differentiate the following functions w.r.t.x:

fraction numerator straight e to the power of straight x squared end exponent space tan to the power of negative 1 end exponent straight x over denominator square root of 1 plus straight x squared end root end fraction

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417. Differentiate the following functions w.r.t.x:

fraction numerator cos to the power of negative 1 end exponent begin display style straight x over 2 end style over denominator square root of 2 straight x plus 7 end root end fraction comma space minus 2 less than straight x less than 2

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418. Differentiate :

fraction numerator straight x space cos to the power of negative 1 end exponent straight x over denominator square root of 1 minus straight x squared end root end fraction comma space straight x not equal to 1 comma space straight w. straight r. straight t. straight x

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419.

Prove space that space cot to the power of negative 1 end exponent straight x space plus space cot minus 1 space 1 over straight x space is space constant.

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420. $If space straight y equals fraction numerator 2 over denominator square root of straight a squared minus straight b squared end root end fraction tan to the power of negative 1 end exponent open parentheses square root of fraction numerator straight a minus straight b over denominator straight a plus straight b end fraction end root tan straight x over 2 close parentheses comma space prove space that space dy over dx equals fraction numerator 1 over denominator straight a plus straight b space cos space straight x end fraction comma space straight a greater than straight b greater than 0.$

Here space space straight y equals fraction numerator 2 over denominator square root of straight a squared minus straight b squared end root end fraction tan to the power of negative 1 end exponent open parentheses square root of fraction numerator straight a minus straight b over denominator straight a plus straight b end fraction end root tan straight x over 2 close parentheses therefore space dy over dx equals fraction numerator 2 over denominator square root of straight a squared minus straight b squared end root end fraction. fraction numerator 1 over denominator 1 plus begin display style fraction numerator straight a minus straight b over denominator straight a plus straight b end fraction end style tan squared begin display style straight x over 2 end style end fraction. straight d over dx open parentheses square root of fraction numerator straight a minus straight b over denominator straight a plus straight b end fraction end root tan straight x over 2 close parentheses space space space space space space space space space space space space space equals fraction numerator 2 over denominator square root of straight a squared minus straight b squared end root end fraction. fraction numerator straight a plus straight b over denominator straight a plus straight b plus left parenthesis straight a minus straight b right parenthesis space tan squared begin display style straight x over 2 end style end fraction. square root of fraction numerator straight a minus straight b over denominator straight a plus straight b end fraction end root. sec squared straight x over 2.1 half space space space space space space space space space space space space space equals fraction numerator 2 over denominator square root of straight a squared minus straight b squared end root end fraction cross times fraction numerator straight a plus straight b over denominator 2 end fraction cross times fraction numerator square root of straight a minus straight b end root over denominator square root of straight a plus straight b end root end fraction. fraction numerator sec squared straight x over 2 over denominator straight a open parentheses 1 plus tan squared straight x over 2 close parentheses plus straight b open parentheses 1 minus tan squared straight x over 2 close parentheses end fraction space space space space space space space space space space space space space equals fraction numerator 1 over denominator straight a plus straight b end fraction cross times fraction numerator straight a plus straight b over denominator 1 end fraction. fraction numerator begin display style fraction numerator sec squared straight x over 2 over denominator open parentheses 1 plus tan squared straight x over 2 close parentheses end fraction end style over denominator straight a plus straight b open parentheses begin display style fraction numerator 1 minus tan squared straight x over 2 over denominator 1 plus tan squared straight x over 2 end fraction end style close parentheses end fraction equals fraction numerator begin display style fraction numerator sec squared straight x over 2 over denominator sec squared straight x over 2 end fraction end style over denominator straight a plus straight b space cos space straight x end fraction equals fraction numerator 1 over denominator straight a plus straight b space cos space straight x end fraction

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