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chetibaby
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Tyro
Asked: 2021-01-10T23:29:29+00:00 In:

Find \dfrac{dy}{dx}, if x^{3}-y^{3}=3xy

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Find \dfrac{dy}{dx}, if x^{3}-y^{3}=3xy

calculusimplicit function

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2 Answers

  2. Sengemegz
    Moderator

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

  4. acedstud
    Best Answer
    This answer was edited.

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

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

    1. \frac{d}{dx}(x^3) = 3x^2.
    2. \frac{d}{dx}(y^3) is a bit more involved because it’s a composite function. You can use the chain rule, which states that \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x). In this case, f(u) = u^3 and g(x) = y(x). So, applying the chain rule:

    \frac{d}{dx}(y^3) = 3y^2 \cdot \frac{dy}{dx}.

    Now, differentiate the right side of the equation:

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

    Now, you can rewrite the equation with these derivatives:

    3x^2 - 3y^2 \cdot \frac{dy}{dx} = 3x\frac{dy}{dx} + 3y.

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

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

    Factor out \frac{dy}{dx}:

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

    Now, solve for \frac{dy}{dx} by dividing both sides by 3x + 3y^2:

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

    Simplify the expression:

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

    So, \frac{dy}{dx} = \frac{x^2 - y}{x + y^2} if x^3 - y^3 = 3xy.

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