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Math Help - Find the derivative!

  1. #1
    Affinity
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    Find the derivative!

    Find the derivative of by product rule and quotient rule. (Thanks in advance!)
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  2. #2
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    Quote Originally Posted by Affinity View Post
    Find the derivative of by product rule and quotient rule. (Thanks in advance!)
    Define f(x) = x - 3x \sqrt{x} and g(x) = \sqrt{x}

    Then f'(x) = 1 - 3 \sqrt{x} - 3x \cdot \frac{1}{2} \frac{1}{\sqrt{x}}
    and
    g'(x) = \frac{1}{2} \frac{1}{\sqrt{x}}

    Quotient Rule:
    So h(x) = \frac{f(x)}{g(x)}

    h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}

    h'(x) = \frac{ \left ( 1 - 3 \sqrt{x} - 3x \cdot \frac{1}{2} \frac{1}{\sqrt{x}} \right ) \sqrt{x} - (x - 3x \sqrt{x}) \frac{1}{2} \frac{1}{\sqrt{x}} }{x}

    h'(x) = ...= \frac{\sqrt{x} - 6x}{2x}
    ================================================== ========

    Define f(x) = x - 3x \sqrt{x} and g(x) = \frac{1}{\sqrt{x}}

    Then f'(x) = 1 - 3 \sqrt{x} - 3x \cdot \frac{1}{2} \frac{1}{\sqrt{x}}
    and
    g'(x) = -\frac{1}{2} \frac{1}{\sqrt{x^3}} = -\frac{1}{2x\sqrt{x}}


    Product rule:
    So h(x) = f(x)g(x)

    h'(x) = f'(x)g(x) + f(x)g(x)

    h'(x) = \left (1 - 3 \sqrt{x} - 3x \cdot \frac{1}{2} \frac{1}{\sqrt{x}} \right ) \frac{1}{\sqrt{x}} + (x - 3x \sqrt{x}) \left ( -\frac{1}{2x\sqrt{x}} \right )

    h'(x) = ... = \frac{1}{2 \sqrt{x}} - 3 = \frac{\sqrt{x} - 6}{2x}

    -Dan
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  3. #3
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    Hello, Affinity!

    Differentiate: . f(x) \:=\:\frac{x - 3x\sqrt{x}}{\sqrt{x}}
    by the product rule and quotient rule.

    Product Rule
    We have: . f(x)\;=\;x^{-\frac{1}{2}}\left(x - 3x^{\frac{3}{2}}\right)

    Then: . f'(x)\;=\;x^{-\frac{1}{2}}\left(1 - \frac{9}{2}x^{\frac{1}{2}}\right) + \left(-\frac{1}{2}x^{-\frac{3}{2}}\right)\left(x - 3x^{\frac{3}{2}}\right)

    . . . . . f'(x) \;= \;x^{-\frac{1}{2}} - \frac{9}{2} - \frac{1}{2}x^{-\frac{1}{2}} + \frac{3}{2}

    . . . . . \boxed{f'(x) \;= \;\frac{1}{2}x^{-\frac{1}{2}} - 3}


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

    Then: . f'(x) \;= \;\frac{x^{\frac{1}{2}}\left(1 - \frac{9}{2}x^{\frac{1}{2}}\right) - \left(\frac{1}{2}x^{-\frac{1}{2}}\right)\left(x - 3x^{\frac{3}{2}}\right)}{\left(x^{\frac{1}{2}}\rig  ht)^2}

    . . . . . f'(x) \;= \;\frac{x^{\frac{1}{2}} - \frac{9}{2}x - \frac{1}{2}x^{\frac{1}{2}} + \frac{3}{2}x}{x}

    . . . . . f'(x) \;= \;\frac{\frac{1}{2}x^{\frac{1}{2}} - 3x}{x}

    . . . . . \boxed{f'(x) \;= \;\frac{x^{\frac{1}{2}} - 6x}{2x}}

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