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Multivariable-Calculus · Partial-Derivatives

Gradient of x^2y - y^3/3

x^{2} y - \frac{y^{3}}{3}

Step by step

  1. f(x, y) = x^{2} y - \frac{y^{3}}{3}

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

  2. \frac{\partial f}{\partial x} = 2 x y

    Treat every variable except x as a constant.

  3. \frac{\partial f}{\partial y} = x^{2} - y^{2}

    Treat every variable except y as a constant.

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

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

  5. (0, 0)

    Critical points: where every partial derivative is zero.

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