Home> Questions>Q 3074
Next
Answered

AcedStudy Latest Questions

chetibaby
  • 0
Tyro
Asked: 2021-01-10T23:31:30+00:00 In:

Find \dfrac{dy}{dx}, if x^{2}+y^{2}-2x+y=6

  • 0

Find \dfrac{dy}{dx}, if x^{2}+y^{2}-2x+y=6

calculusimplicit function

You must login to add an answer.

Need An Account,

2 Answers

  2. acedstud
    Best Answer

    To find \frac{dy}{dx} given x^2 + y^2 - 2x + y = 6, 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^2 + y^2 - 2x + y) = \frac{d}{dx}(6).

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

    1. \frac{d}{dx}(x^2) = 2x.
    2. \frac{d}{dx}(y^2) 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^2 and g(x) = y(x). So, applying the chain rule:

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

    3. \frac{d}{dx}(-2x) = -2.

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

    Now, differentiate the right side of the equation:

    \frac{d}{dx}(6) = 0 because the derivative of a constant is 0.

    Now, you can rewrite the equation with these derivatives:

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

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

    2x - 2 + \frac{dy}{dx}(2y + 1) = 0.

    Factor out \frac{dy}{dx}:

    \frac{dy}{dx}(2y + 1) = 2 - 2x.

    Now, solve for \frac{dy}{dx} by dividing both sides by 2y + 1:

    \frac{dy}{dx} = \frac{2 - 2x}{2y + 1}.

    So, \frac{dy}{dx} = \frac{2 - 2x}{2y + 1} is the derivative with respect to x of the equation x^2 + y^2 - 2x + y = 6.

  4. Sengemegz
    Moderator

    x^{2}+y^{2}-2x+y=6
    \Rightarrow 2x+2y\dfrac{dy}{dx}-2+\dfrac{dy}{dx}=0
    \Rightarrow 2y\dfrac{dy}{dx}+\dfrac{dy}{dx}=-2x+2
    \Rightarrow \dfrac{dy}{dx}(2y+1)=-2x+2
    \therefore \dfrac{dy}{dx}=\dfrac{-2x+2}{2y+1}

Related Questions