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chetibaby
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Tyro
Asked: 2021-01-10T23:34:02+00:00 In:

Find \dfrac{dy}{dx}, if 3xy+x^{2}=7y

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Find \dfrac{dy}{dx}, if 3xy+x^{2}=7y

calculusimplicit function

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2 Answers

  2. Sengemegz
    Moderator

    3xy+x^{2}=7y
    \Rightarrow 3x\dfrac{dy}{dx}+3y+2x=7\dfrac{dy}{dx}
    \Rightarrow 3x\dfrac{dy}{dx}-7\dfrac{dy}{dx}=-2x-3y
    \Rightarrow \dfrac{dy}{dx}(3x-7)=-2x-3y
    \therefore \dfrac{dy}{dx}=\dfrac{-2x-3y}{3x-7}

  4. acedstud
    Best Answer

    To find \frac{dy}{dx} given 3xy + x^2 = 7y, you can use implicit differentiation. The equation is not given in the form y = f(x), so we will differentiate both sides of the equation with respect to x using implicit differentiation.

    Start by differentiating both sides of the equation:

    \frac{d}{dx}(3xy + x^2) = \frac{d}{dx}(7y).

    Now, differentiate each term on the left side of the equation separately:

    1. \frac{d}{dx}(3xy) = 3x\frac{dy}{dx} + 3y.
    2. \frac{d}{dx}(x^2) = 2x.

    Now, differentiate the right side of the equation:

    \frac{d}{dx}(7y) = 7\frac{dy}{dx}.

    Now, you can rewrite the equation with these derivatives:

    3x\frac{dy}{dx} + 3y + 2x = 7\frac{dy}{dx}.

    Now, isolate the term with \frac{dy}{dx}:

    3x\frac{dy}{dx} - 7\frac{dy}{dx} = -3y - 2x.

    Factor out \frac{dy}{dx}:

    (3x - 7)\frac{dy}{dx} = -3y - 2x.

    Now, solve for \frac{dy}{dx} by dividing both sides by (3x - 7):

    \frac{dy}{dx} = \frac{-3y - 2x}{3x - 7}.

    So, \frac{dy}{dx} = \frac{-3y - 2x}{3x - 7} is the derivative with respect to x of the equation 3xy + x^2 = 7y.

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