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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.

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

Let space straight y equals 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 comma space straight u equals square root of straight x space dy over dx equals fraction numerator 1 over denominator square root of 1 minus open parentheses fraction numerator 1 minus straight x over denominator 1 plus straight x end fraction close parentheses squared end root end fraction. straight d over dx open parentheses fraction numerator 1 minus straight x over denominator 1 plus straight x end fraction close parentheses equals fraction numerator 1 plus straight x over denominator square root of left parenthesis 1 plus straight x right parenthesis squared minus left parenthesis 1 minus straight x right parenthesis squared end root end fraction cross times fraction numerator left parenthesis 1 plus straight x right parenthesis left parenthesis negative 1 right parenthesis minus left parenthesis 1 minus straight x right parenthesis left parenthesis 1 right parenthesis over denominator left parenthesis 1 plus straight x right parenthesis squared end fraction space space space space space space space space space equals fraction numerator 1 plus straight x over denominator square root of 4 straight x end root end fraction cross times fraction numerator negative 2 over denominator left parenthesis 1 plus straight x right parenthesis squared end fraction equals negative fraction numerator 1 over denominator square root of straight x left parenthesis 1 plus straight x right parenthesis end fraction space du over dx equals fraction numerator 1 over denominator 2 square root of straight x end fraction 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 negative fraction numerator 1 over denominator square root of straight x left parenthesis 1 plus straight x right parenthesis end fraction cross times fraction numerator 2 square root of straight x over denominator 1 end fraction equals negative fraction numerator 2 over denominator 1 plus straight x end fraction

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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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