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Multivariable-Calculus · Partial-Derivatives

Gradient of x^2 + y^2 + z^2

x^{2} + y^{2} + z^{2}

Inyathelo ngenyathelo

  1. f(x, y, z) = x^{2} + y^{2} + z^{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} = 2 x

    Treat every variable except x as a constant.

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

    Treat every variable except y as a constant.

  4. \frac{\partial f}{\partial z} = 2 z

    Treat every variable except z as a constant.

  5. \nabla f = \left[\begin{matrix}2 x\\2 y\\2 z\end{matrix}\right]

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

  6. (0, 0, 0)

    Critical points: where every partial derivative is zero.

Bonisa impendulo
\nabla f = \left[\begin{matrix}2 x\\2 y\\2 z\end{matrix}\right]