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chetibaby
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Tyro
Asked: 2021-01-10T23:32:51+00:00 In:

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

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

calculusimplicit function

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2 Answers

  2. Sengemegz
    Moderator

    xy+x^{2}=3
    \Rightarrow x\dfrac{dy}{dx}+y+2x=0
    \Rightarrow x\dfrac{dy}{dx}=-2x-y
    \therefore \dfrac{dy}{dx}=\dfrac{-2x-y}{x}

  4. acedstud
    Best Answer

    To find \frac{dy}{dx} given xy + x^2 = 3, 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}(xy + x^2) = \frac{d}{dx}(3).

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

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

    Now, differentiate the right side of the equation:

    \frac{d}{dx}(3) = 0 because the derivative of a constant is 0.

    Now, you can rewrite the equation with these derivatives:

    x\frac{dy}{dx} + y + 2x = 0.

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

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

    Now, solve for \frac{dy}{dx} by dividing both sides by x:

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

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

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