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Continuity and Differentiability

Short Answer Type

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

cot to the power of negative 1 end exponent left parenthesis cosec space straight x plus cot space straight x right parenthesis

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

cos to the power of negative 1 end exponent left parenthesis sin space straight x right parenthesis

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443.

Find space dy over dx comma space when space straight y equals sin to the power of negative 1 end exponent straight x plus sin to the power of negative 1 end exponent square root of 1 minus straight x squared end root.

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444.

Differentiate space straight x to the power of straight x sin to the power of negative 1 end exponent square root of straight x space straight w. straight r. straight t. straight x.

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445.

If space straight y equals sin left parenthesis 2 space sin to the power of negative 1 end exponent straight x right parenthesis comma space show space that space dy over dx equals 2 square root of fraction numerator 1 minus straight y squared over denominator 1 minus straight x squared end fraction end root.

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446.

Find space dy over dx space of space the space following space 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 comma space 0 less than straight x less than 1 space straight w. straight r. straight t. straight x. space

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447. $Differentiate space tan to the power of negative 1 end exponent open parentheses fraction numerator square root of 1 plus straight x squared end root minus 1 over denominator straight x end fraction close parentheses comma 0 less than straight x less than 1 comma space straight w. straight r. straight t. straight x.$

Let space straight y equals tan to the power of negative 1 end exponent open parentheses fraction numerator square root of 1 plus straight x squared end root minus 1 over denominator straight x end fraction close parentheses Put space straight x equals tan space straight theta therefore space space space straight y equals tan to the power of negative 1 end exponent open square brackets fraction numerator square root of 1 plus straight x squared end root minus 1 over denominator tan space straight theta end fraction close square brackets equals tan to the power of negative 1 end exponent open square brackets fraction numerator square root of sec squared space straight theta end root minus 1 over denominator tan space straight theta end fraction close square brackets equals tan to the power of negative 1 end exponent open square brackets fraction numerator sec space straight theta minus 1 over denominator tan space straight theta end fraction close square brackets space space space space space space space space space equals tan to the power of negative 1 end exponent open square brackets fraction numerator begin display style fraction numerator 1 over denominator cos space straight theta end fraction end style minus 1 over denominator begin display style fraction numerator sin space straight theta over denominator cos space straight theta end fraction end style end fraction close square brackets equals tan to the power of negative 1 end exponent open square brackets fraction numerator 1 minus cos space straight theta over denominator sin space straight theta end fraction close square brackets equals tan to the power of negative 1 end exponent open square brackets fraction numerator 2 sin squared begin display style straight theta over 2 end style over denominator 2 sin squared begin display style straight theta over 2 end style cos space begin display style straight theta over 2 end style end fraction close square brackets space space space space space space space space space equals tan to the power of negative 1 end exponent open square brackets fraction numerator sin space begin display style straight theta over 2 end style over denominator cos space begin display style straight theta over 2 end style end fraction close square brackets equals tan to the power of negative 1 end exponent open square brackets tan space straight theta over 2 close square brackets equals straight theta over 2 therefore space space space straight y equals 1 half tan to the power of negative 1 end exponent straight x therefore space dy over dx equals 1 half cross times fraction numerator 1 over denominator 1 plus straight x squared end fraction equals fraction numerator 1 over denominator 2 left parenthesis 1 plus straight x squared right parenthesis end fraction

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448.

Find space dy over dx space in space the space following space colon space 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

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449.

Find space fraction numerator dy over denominator dx space end fraction space in space the space following space space tan to the power of negative 1 end exponent open parentheses fraction numerator 2 straight x over denominator 1 minus straight x squared end fraction close parentheses

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450.

Find space dy over dx space in space the space following space colon space tan to the power of negative 1 end exponent open parentheses fraction numerator 3 straight x minus straight x cubed over denominator 1 minus 3 straight x squared end fraction close parentheses comma space minus fraction numerator 1 over denominator square root of 3 end fraction less than straight x less than fraction numerator 1 over denominator square root of 3 end fraction

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