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Continuity and Differentiability

Short Answer Type

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

Let space straight y equals 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 semicolon space Put space straight a space straight x equals tan space straight theta space space space space space space space space space equals tan to the power of negative 1 end exponent open square brackets fraction numerator square root of 1 plus tan squared space straight theta end root minus 1 over denominator tan space straight theta end fraction close square brackets equals tan to the power of negative 1 end exponent open square brackets fraction numerator sec space straight theta minus 1 over denominator tan space straight theta end fraction close square brackets space space space space space space space space space equals tan to the power of negative 1 end exponent open square brackets fraction numerator begin display style fraction numerator 1 over denominator cos space straight theta end fraction end style minus 1 over denominator begin display style fraction numerator sin space straight theta over denominator cos space straight theta end fraction end style end fraction close square brackets equals equals tan to the power of negative 1 end exponent open square brackets fraction numerator 1 minus cos space straight theta over denominator sin space straight theta end fraction close square brackets space space space space space space space space space equals tan to the power of negative 1 end exponent open square brackets fraction numerator 2 space sin squared begin display style straight theta over 2 end style over denominator 2 space sin begin display style straight theta over 2 end style cos begin display style straight theta over 2 end style end fraction close square brackets equals tan to the power of negative 1 end exponent open square brackets tan space straight theta over 2 close square brackets equals straight theta over 2 space space space space space space space straight y equals 1 half tan to the power of negative 1 end exponent straight x therefore space dy over dx equals 1 half. fraction numerator straight a over denominator 1 plus left parenthesis straight a space straight x right parenthesis squared end fraction equals fraction numerator straight a over denominator 2 left parenthesis 1 plus straight a squared space straight x squared right parenthesis end fraction

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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

93 Views

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

90 Views

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

78 Views

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

88 Views

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

73 Views