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

Let space space space space space straight y equals 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 therefore space dy over dx equals fraction numerator square root of 1 minus straight x squared end root. begin display style straight d over dx end style left parenthesis straight x space cos to the power of negative 1 end exponent straight x right parenthesis minus left parenthesis straight x space cos to the power of negative 1 end exponent straight x right parenthesis. begin display style straight d over dx end style open parentheses square root of 1 minus straight x squared end root close parentheses over denominator open parentheses square root of 1 minus straight x squared end root close parentheses squared end fraction space space space space space space space space space space space space space equals fraction numerator square root of 1 minus straight x squared end root open parentheses straight x. begin display style fraction numerator negative 1 over denominator square root of 1 minus straight x squared end root end fraction end style plus space cos to the power of negative 1 end exponent straight x.1 close parentheses minus left parenthesis straight x space cos to the power of negative 1 end exponent straight x right parenthesis. begin display style fraction numerator negative 2 straight x over denominator 2 square root of 1 minus straight x squared end root end fraction end style over denominator 1 minus straight x squared end fraction space space space space space space space space space space space space space equals fraction numerator negative straight x plus square root of 1 minus straight x squared end root space cos to the power of negative 1 end exponent straight x plus begin display style fraction numerator straight x squared 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 end style over denominator 1 minus straight x squared end fraction space space space space space space space space space space space space space equals fraction numerator negative straight x square root of 1 minus straight x squared end root space plus left parenthesis 1 minus straight x squared right parenthesis cos to the power of negative 1 end exponent straight x plus straight x squared cos to the power of negative 1 end exponent straight x over denominator open parentheses 1 minus straight x squared close parentheses to the power of begin display style 3 over 2 end style end exponent end fraction space space space space space space space space space space space space space equals fraction numerator negative straight x square root of 1 minus straight x squared end root space plus cos to the power of negative 1 end exponent space straight x minus straight x squared cos to the power of negative 1 end exponent straight x plus straight x squared cos to the power of negative 1 end exponent straight x over denominator open parentheses 1 minus straight x squared close parentheses to the power of begin display style 3 over 2 end style end exponent end fraction therefore space dy over dx equals fraction numerator negative straight x square root of 1 minus straight x squared end root space plus cos to the power of negative 1 end exponent space straight x over denominator open parentheses 1 minus straight x squared close parentheses to the power of 3 over 2 end exponent end fraction

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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.

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