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Multivariable-Calculus · Partial-Derivatives

Gradient of y - y/(x^2 + y^2)

y - \frac{y}{x^{2} + y^{2}}

Step by step

  1. f(x, y) = y - \frac{y}{x^{2} + y^{2}}

    The gradient is the vector of partial derivatives. Differentiate with respect to each variable, holding the others constant.

  2. \frac{\partial f}{\partial x} = \frac{2 x y}{\left(x^{2} + y^{2}\right)^{2}}

    Treat every variable except x as a constant.

  3. \frac{\partial f}{\partial y} = \frac{2 y^{2}}{\left(x^{2} + y^{2}\right)^{2}} + 1 - \frac{1}{x^{2} + y^{2}}

    Treat every variable except y as a constant.

  4. \nabla f = \left[\begin{matrix}\frac{2 x y}{\left(x^{2} + y^{2}\right)^{2}}\\\frac{2 y^{2}}{\left(x^{2} + y^{2}\right)^{2}} + 1 - \frac{1}{x^{2} + y^{2}}\end{matrix}\right]

    Assemble the gradient vector. It points in the direction of steepest ascent.

  5. (-1, 0),\ (1, 0)

    Critical points: where every partial derivative is zero.

Reveal the answer
\nabla f = \left[\begin{matrix}\frac{2 x y}{\left(x^{2} + y^{2}\right)^{2}}\\\frac{2 y^{2}}{\left(x^{2} + y^{2}\right)^{2}} + 1 - \frac{1}{x^{2} + y^{2}}\end{matrix}\right]