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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. $If space straight y equals sin to the power of negative 1 end exponent open parentheses fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction close parentheses plus sec to the power of negative 1 end exponent open parentheses fraction numerator 1 plus straight x squared over denominator 1 minus straight x squared end fraction close parentheses comma space prove space that space dy over dx equals fraction numerator 4 over denominator 1 plus straight x squared end fraction$

Here space 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 Put space space space space space straight x equals tan space straight theta therefore space space space space If space straight y equals sin to the power of negative 1 end exponent open parentheses fraction numerator 2 tan space straight theta over denominator 1 plus tan squared space straight theta end fraction close parentheses plus sec to the power of negative 1 end exponent open parentheses fraction numerator 1 plus tan squared space straight theta over denominator 1 minus tan squared space straight theta end fraction close parentheses or space space space space space space space space straight y equals sin to the power of negative 1 end exponent left parenthesis sin space 2 straight theta right parenthesis plus sec to the power of negative 1 end exponent open parentheses fraction numerator 1 plus tan squared space straight theta over denominator 1 minus tan squared straight theta end fraction close parentheses therefore space space space space space space space space straight y equals 2 space straight theta plus 2 space straight theta comma space or space straight y equals 4 space straight theta therefore space space space space space space space space straight y equals 4 space tan to the power of negative 1 end exponent straight x therefore 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

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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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