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Multivariable-Calculus · Partial-Derivatives

Gradient of x·y

x y

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  1. f(x, y) = x y

    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

    Treat every variable except x as a constant.

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

    Treat every variable except y as a constant.

  4. \nabla f = \left[\begin{matrix}y\\x\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.

@ action
\nabla f = \left[\begin{matrix}y\\x\end{matrix}\right]