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Math Help - Derivatives help

  1. #1
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    Derivatives help

    Hi, I'm having difficulty solving and simplifying these two derivatives:

    y = x
    _____
    √(x^2-1)

    and

    y= (3x^2 - 2)^2
    √(x^2-5)

    Thank you for any help!
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  2. #2
    Eater of Worlds
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    You can try the product rule on the first one.

    x(x^{2}-1)^{\frac{-1}{2}}

    x(\frac{-1}{2})(x^{2}-1)^{\frac{-3}{2}}(2x)+(x^{2}-1)^{\frac{-1}{2}}

    -x^{2}(x^{2}-1)^{\frac{-3}{2}}+(x^{2}-1)^{\frac{-1}{2}}

    Factor:

    (x^{2}-1)^{\frac{-1}{2}}(1-x^{2}(x^{2}-1)^{-1})

    (x^{2}-1)^{\frac{-1}{2}}(\frac{-1}{x^{2}-1})

    \frac{-1}{(x^{2}-1)^{\frac{3}{2}}}
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  3. #3
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    Hello, starswept!

    There are some tricks to simplifying expressions with negative exponents.


    y \;=\;\frac{x}{\sqrt{x^2-1}}
    We have: . y \;=\;\frac{x}{(x^2-1)^{\frac{1}{2}}}

    Quotient Rule: . \frac{dy}{dx} \;=\;\frac{(x^2-1)^{\frac{1}{2}}\cdot1 - x\cdot\frac{1}{2}(x^2-1)^{-\frac{1}{2}}\cdot2x}{\left[(x^2-1)^{\frac{1}{2}}\right]^2}\;=\;\frac{(x^2-1)^{\frac{1}{2}} - x^2(x^2-1)^{-\frac{1}{2}}}{x^2-1}

    Multiply by \frac{(x^2-1)^{\frac{1}{2}}}{(x^2-1)^{\frac{1}{2}}}

    . . \frac{dy}{dx}\;=\;\frac{(x^2-1)^{\frac{1}{2}}}{(x^2-1)^{\frac{1}{2}}}\cdot\frac{(x^2-1)^{\frac{1}{2}} - x^2(x^2-1)^{-\frac{1}{2}}}{x^2-1} \;= \;\frac{x^2 - 1 - x^2}{(x^2-1)^{\frac{3}{2}}}

    Therefore: . \frac{dy}{dx}\;=\;\frac{-1}{(x^2-1)^{\frac{3}{2}}}




    y \;= \;(3x^2 - 2)^2\sqrt{x^2-5}

    We have: . y \;=\;(3x^2-2)^2(x^2-1)^{\frac{1}{2}}

    Produce Rule: . \frac{dy}{dx} \;= \;(3x^2-2)^2\cdot\frac{1}{2}(x^2-5)^{-\frac{1}{2}}\cdot2x \:+ \:(x^2-5)^{\frac{1}{2}}\cdot2(3x^2-2)\cdot6x

    . . . . . . . . . . . \frac{dy}{dx}\;= \;x(3x^2-2)^2(x^2-5)^{-\frac{1}{2}} + 12x(3x^2-2)(x^2-5)^{\frac{1}{2}}

    Factor: . \frac{dy}{dx} \;=\;x(3x^2-2)\,\left[(3x^2-2)(x^2-5)^{-\frac{1}{2}} + 12(x^2-5)^{\frac{1}{2}}\right]

    Multiply by \frac{(x^2-5)^{\frac{1}{2}}}{(x^2-5)^{\frac{1}{2}}}

    . . \frac{dy}{dx} \;= \;x(3x^2-2)\!\cdot\!\frac{(3x^2-2)(x^2-5)^{-\frac{1}{2}} + 12(x^2-5)^{\frac{1}{2}}}{1}\cdot\frac{(x^2-5)^{\frac{1}{2}}}{(x^2-5)^{\frac{1}{2}}}
    .
    . . \frac{dy}{dx} \;=\;x(3x^2-2)\cdot\frac{(3x^2 - 2) + 12(x^2-5)}{(x^2-5)^{\frac{1}{2}}} <br />

    Therefore: . \frac{dy}{dx}\;=\;\frac{x(3x^2-2)(15x^2-62)}{(x^2-5)^{\frac{1}{2}}}

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