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Continuity and Differentiability

Short Answer Type

471.

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

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472.

If space straight y equals 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 plus sec to the power of negative 1 end exponent open parentheses fraction numerator 1 plus straight x squared over denominator 1 minus straight x squared end fraction close parentheses comma space prove space that space dy over dx equals fraction numerator 4 over denominator 1 plus straight x squared end fraction

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473.

If space straight y equals sin open square brackets 2 space tan to the power of negative 1 end exponent square root of fraction numerator 1 minus straight x over denominator 1 plus straight x end fraction end root close square brackets comma space prove space that space dy over dx equals negative fraction numerator straight x over denominator square root of 1 minus straight x squared end root end fraction

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474.

If 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 square root of 1 minus straight x squared end root over denominator square root of 1 plus straight x squared end root plus square root of 1 minus straight x squared end root end fraction close square brackets comma space find space dy over dx

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475. $If space space straight y equals tan to the power of negative 1 end exponent open square brackets fraction numerator square root of 1 plus straight x squared end root plus square root of 1 minus straight x squared end root over denominator square root of 1 plus straight x squared end root minus square root of 1 minus straight x squared end root end fraction close square brackets comma space prove space that space dy over dx equals negative fraction numerator straight x over denominator square root of 1 minus straight x to the power of 4 end root end fraction$

space 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 plus square root of 1 minus straight x squared end root over denominator square root of 1 plus straight x squared end root minus square root of 1 minus straight x squared end root end fraction close square brackets Put space space straight x squared equals cos space straight theta therefore space space straight y equals tan to the power of negative 1 end exponent open square brackets fraction numerator square root of 1 plus cos space straight theta end root plus square root of 1 minus cos space straight theta end root over denominator square root of 1 plus cos space straight theta end root minus square root of 1 minus cos space straight theta end root end fraction close square brackets equals tan space to the power of negative 1 end exponent open square brackets fraction numerator square root of 2 cos squared straight theta over 2 end root plus square root of 2 sin squared straight theta over 2 end root over denominator square root of 2 cos squared straight theta over 2 end root minus square root of 2 sin squared straight theta over 2 end root end fraction close square brackets equals tan to the power of negative 1 end exponent open square brackets fraction numerator square root of 2 space end root cos straight theta over 2 minus square root of 2 space end root sin space straight theta over 2 over denominator square root of 2 space end root cos straight theta over 2 plus square root of 2 space end root sin space straight theta over 2 end fraction close square brackets equals tan to the power of negative 1 end exponent open square brackets fraction numerator cos straight theta over 2 plus sin straight theta over 2 over denominator cos straight theta over 2 minus sin straight theta over 2 end fraction close square brackets equals tan to the power of negative 1 end exponent open square brackets fraction numerator 1 plus tan space straight theta over 2 over denominator 1 minus tan space straight theta over 2 end fraction close square brackets equals tan to the power of negative 1 end exponent open square brackets tan open parentheses straight pi over 4 plus straight theta over 2 close parentheses close square brackets equals straight pi over 4 plus straight theta over 2 therefore space space straight y equals straight pi over 4 plus 1 half cos to the power of negative 1 end exponent open parentheses straight x squared close parentheses therefore space dy over dx equals 0 plus 1 half open square brackets negative fraction numerator 1 over denominator square root of 1 minus straight x to the power of 4 end root end fraction close square brackets straight d over dx left parenthesis straight x squared right parenthesis equals 1 half cross times negative fraction numerator 1 over denominator square root of 1 minus straight x to the power of 4 end root end fraction cross times 2 straight x equals negative fraction numerator straight x over denominator square root of 1 minus straight x to the power of 4 end root end fraction.

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476.

Find the equation of the normal to the curve x² = 4 y which passes through the point (1, 2).

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477. Differentiate sin (tan^-1 2x) w.r.t.x and express the result free from trigonometric symbols.

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478.

If y=sin^-1(cos x)+cos^-1(sin x), find $dy over dx$ and express the result free from trigonometrical symbols.

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479.

If space straight y equals space tan to the power of negative 1 end exponent open parentheses fraction numerator 5 straight a over denominator straight a squared minus 6 straight x squared end fraction close parentheses comma space prove space that space dy over dx equals fraction numerator 3 straight a over denominator straight a squared plus 9 straight x squared end fraction plus fraction numerator 2 straight a over denominator straight a squared plus 4 straight x squared end fraction

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480.

If space straight y equals fraction numerator straight x space sin to the power of negative 1 end exponent straight x over denominator square root of 1 minus straight x squared end root end fraction plus log square root of 1 minus straight x squared end root comma space prove space that space dy over dx equals fraction numerator sin to the power of negative 1 end exponent straight x over denominator left parenthesis 1 minus straight x squared right parenthesis to the power of begin display style 3 over 2 end style end exponent end fraction

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