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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.