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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.

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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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Let space space space space space space space space space straight y equals sin to the power of negative 1 end exponent straight x space comma space straight u equals cos to the power of negative 1 end exponent square root of 1 minus straight x squared end root therefore space space space space space dy over dx equals fraction numerator 1 over denominator square root of 1 minus straight x squared end root end fraction space space space space space space space space space du over dx equals negative fraction numerator 1 over denominator square root of 1 minus left parenthesis 1 minus straight x squared right parenthesis end root end fraction. straight d over dx left parenthesis square root of 1 minus straight x squared end root right parenthesis equals negative 1 over straight x. fraction numerator negative 2 straight x over denominator 2 square root of 1 minus straight x squared end root end fraction equals fraction numerator 1 over denominator square root of 1 minus straight x squared end root 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 1 over denominator square root of 1 minus straight x squared end root end fraction cross times fraction numerator square root of 1 minus straight x squared end root over denominator 1 end fraction equals 1

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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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