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Continuity and Differentiability

Short Answer Type

491. $Differentiate space tan to the power of negative 1 end exponent open parentheses fraction numerator 2 straight x over denominator 1 minus straight x squared end fraction close parentheses space straight w. straight r. straight t. space sin to the power of negative 1 end exponent open parentheses fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction close parentheses.$

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

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492.

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

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493. Prove that the derivative of tan^-1

open parentheses fraction numerator square root of 1 plus straight x squared end root minus 1 over denominator straight x end fraction close parentheses

w.r.t. tan^-1 x is independent of x.

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494.

Find space the space derivative space of space tan to the power of negative 1 end exponent fraction numerator 2 straight x over denominator 1 minus straight x squared end fraction space straight w. straight r. straight t. space tan to the power of negative 1 end exponent straight x

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495.

Differentiate space sec to the power of negative 1 end exponent open parentheses fraction numerator 1 over denominator 2 straight x squared minus 1 end fraction close parentheses space straight w. straight r. straight t. space sin to the power of negative 1 end exponent left parenthesis 3 straight x minus 4 straight x cubed right parenthesis.

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496.

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

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497.

Differentiate space sin to the power of negative 1 end exponent open parentheses fraction numerator 1 minus straight x over denominator 1 plus straight x end fraction close parentheses space straight w. straight r. straight t. space square root of straight x

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498.

Differentiate space cos to the power of negative 1 end exponent straight theta space straight w. straight r. straight t. space log left parenthesis 1 plus straight theta right parenthesis

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499.

Differentiate space sin to the power of negative 1 end exponent straight theta space straight w. straight r. straight t. space log left parenthesis 1 plus straight theta right parenthesis

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500.

Differentiate space log left parenthesis 1 plus straight theta right parenthesis space straight w. straight r. straight t. space sin to the power of negative 1 end exponent space straight theta

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