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Math Help - Directional derivatives

  1. #1
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    Directional derivatives

    Find the directional derivative of f(x,y) = \sqrt{xy} at P(2,8) in the direction of Q(5,4)?

    I'm not sure exactly what to do.

    I have differentiated the equation, which gives me

     f_x(x,y) = \frac{y}{2 \sqrt{xy}}, \ \ \mbox{and} \ \ f_y(x,y) = \frac{x}{2 \sqrt{xy}} with  \Delta x = 3, \Delta y = -4 and from this point I have no clue on what to do.
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  2. #2
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    Quote Originally Posted by lllll View Post
    Find the directional derivative of f(x,y) = \sqrt{xy} at P(2,8) in the direction of Q(5,4)?

    I'm not sure exactly what to do.

    I have differentiated the equation, which gives me

     f_x(x,y) = \frac{y}{2 \sqrt{xy}}, \ \ \mbox{and} \ \ f_y(x,y) = \frac{x}{2 \sqrt{xy}} with  \Delta x = 3, \Delta y = -4 and from this point I have no clue on what to do.
    The directional derivative is given by the formula:

    \frac{df}{d\vec{l}} = \nabla f \cdot \vec{\hat{l}}

    where \vec{\hat{l}} is a unit vector in the given direction.

    In your case \vec{\hat{l}} = \frac{5 i + 4 j}{\sqrt{41}} and \nabla f = \frac{y}{2 \sqrt{xy}} i +  \frac{x}{2 \sqrt{xy}} j.

    Now substitute the given point.
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