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Math Help - Implicit Differentiation & Horizontal tangents

  1. #1
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    Implicit Differentiation & Horizontal tangents

    Given the function 5x^2+8xy+5y^2 = 45 find the derivative and horizontal tangent(s).


    I've obtained the derivative as
    dy/dx = (-5x-4y)/(4x+5y)

    However i am confused about the horizontal tangent, i understand that a tangent is horizontal when dy/dx = 0, but i am unsure of what to do next.

    0 = (-5x-4y)/(4x+5y)

    I'm not after the answer, but the next step(s) required.
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  2. #2
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    It's tough to supply a next step when you already have.

    Find where the numerator is zero and the denominator is not.

    You may wish to try it after completing the square.

    5\cdot\left(x+y\right)^{2} - 2xy = 45

    Maybe not. It was just a thought.

    Anyway, since there is nowhere but the Origin where both numerator and denominator are zero, just solve the numerator. Obviously, you will get a line, not a point. Substitute into the original equation and get the two points.
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  3. #3
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    Quote Originally Posted by JohnnyB View Post
    Given the function 5x^2+8xy+5y^2 = 45 find the derivative and horizontal tangent(s).


    I've obtained the derivative as
    dy/dx = (-5x-4y)/(4x+5y)

    However i am confused about the horizontal tangent, i understand that a tangent is horizontal when dy/dx = 0, but i am unsure of what to do next.

    0 = (-5x-4y)/(4x+5y)

    I'm not after the answer, but the next step(s) required.
    if \frac{dy}{dx} = 0 , then -5x-4y = 0 ... y = -\frac{5x}{4}

    substitute \left(-\frac{5x}{4}\right) for y in the equation for the curve and solve for x ... then determine the corresponding y-value for each x solution you find.
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  4. #4
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    Probably a stupid question...

    but why is the numerator of the derivative used and not the denominator, or both?
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  5. #5
    Super Member bigwave's Avatar
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    Quote Originally Posted by JohnnyB View Post
    Probably a stupid question...

    but why is the numerator of the derivative used and not the denominator, or both?
    if the denominator is zero it is undefined
    if the numerator is zero you have zero for an answer
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