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Multivariable-Calculus · Partial-Derivatives

Gradient of x·y + 1/y + 1/x

x y + \frac{1}{y} + \frac{1}{x}

tallaabo tallaabo

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

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

  2. \frac{\partial f}{\partial x} = y - \frac{1}{x^{2}}

    Treat every variable except x as a constant.

  3. \frac{\partial f}{\partial y} = x - \frac{1}{y^{2}}

    Treat every variable except y as a constant.

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

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

  5. (1, 1)

    Critical points: where every partial derivative is zero.

Jawaabta muuji
\nabla f = \left[\begin{matrix}y - \frac{1}{x^{2}}\\x - \frac{1}{y^{2}}\end{matrix}\right]