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chetibaby
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Tyro
Asked: 2021-01-10T23:27:53+00:00 In:

Find \dfrac{dy}{dx}, if x^{2}y-5x=3

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Find \dfrac{dy}{dx}, if x^{2}y-5x=3

calculusimplicit function

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2 Answers

  2. Sengemegz
    Moderator

    x^{2}y-5x=3
    \Rightarrow 2xy+x^{2}\dfrac{dy}{dx}-5=0
    \Rightarrow x^{2}\dfrac{dy}{dx}=5-2xy
    \therefore \dfrac{dy}{dx}=\dfrac{5-2xy}{x^{2}}

  4. acedstud
    Best Answer
    This answer was edited.

    To find \frac{dy}{dx} given x^2y - 5x = 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:

        \[\frac{d}{dx}(x^2y - 5x) = \frac{d}{dx}(3)\]

    which can be further simplified as:

        \[\frac{d}{dx}(x^2y) - \frac{d}{dx}(5x) = 0\]

    (the derivative of a constant, 3, is 0).

    Now, apply the product rule to the first term of the equation. Recall that the product rule states:

        \[\frac{d}{dx}(u(x)v(x)) = u(x)\frac{dv}{dx} + v(x)\frac{du}{dx}\]

    In our case, let u(x) = x^2 and v(x) = y. We also need to differentiate y with respect to x, which is \frac{dy}{dx}.

    Now, we can differentiate each term, applying the product rule to the first term:

    1. \frac{d}{dx}(x^2y) = x^2\frac{dy}{dx} + 2xy,
    2. \frac{d}{dx}(5x) = 5.

    Now, substitute these results back into the equation:

        \[x^2\frac{dy}{dx} + 2xy - 5 = 0\]

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

        \[x^2\frac{dy}{dx} = 5 - 2xy\]

    Finally, divide by x^2 to solve for \frac{dy}{dx}:

        \[\frac{dy}{dx} = \frac{5 - 2xy}{x^2}\]

    So, \frac{dy}{dx} = \frac{5 - 2xy}{x^2} if x^2y - 5x = 3.

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