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

Here space space space space space space straight y equals sin left parenthesis 2 space sin to the power of negative 1 end exponent straight x right parenthesis Put space sin to the power of negative 1 end exponent straight x equals straight theta space straight i. straight e. space straight x equals sin space straight theta therefore space space space space space space space space space space space straight y equals sin space 2 straight theta space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis therefore space space space space space dy over dx equals 2 space cos space 2 straight theta equals 2 square root of 1 minus sin squared space straight theta end root therefore space space space space space dy over dx equals 2 square root of 1 minus straight y squared end root space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left square bracket because space of space left parenthesis 1 right parenthesis right square bracket Now space dy over dx equals dy over dθ cross times dθ over dx space space space space space space space space space space space space space space space space space equals 2 square root of 1 minus straight y squared end root cross times fraction numerator 1 over denominator square root of 1 minus straight x squared end root end fraction comma space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left square bracket because space straight theta equals sin to the power of negative 1 end exponent straight x right square bracket therefore space space space space space 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.

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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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