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Multivariable-Calculus · Partial-Derivatives

Gradient of 1/sqrt(x^2 + y^2 + z^2)

\frac{1}{\sqrt{x^{2} + y^{2} + z^{2}}}

Step by step

  1. f(x, y, z) = \frac{1}{\sqrt{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} = - \frac{x}{\left(x^{2} + y^{2} + z^{2}\right)^{\frac{3}{2}}}

    Treat every variable except x as a constant.

  3. \frac{\partial f}{\partial y} = - \frac{y}{\left(x^{2} + y^{2} + z^{2}\right)^{\frac{3}{2}}}

    Treat every variable except y as a constant.

  4. \frac{\partial f}{\partial z} = - \frac{z}{\left(x^{2} + y^{2} + z^{2}\right)^{\frac{3}{2}}}

    Treat every variable except z as a constant.

  5. \nabla f = \left[\begin{matrix}- \frac{x}{\left(x^{2} + y^{2} + z^{2}\right)^{\frac{3}{2}}}\\- \frac{y}{\left(x^{2} + y^{2} + z^{2}\right)^{\frac{3}{2}}}\\- \frac{z}{\left(x^{2} + y^{2} + z^{2}\right)^{\frac{3}{2}}}\end{matrix}\right]

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

Reveal the answer
\nabla f = \left[\begin{matrix}- \frac{x}{\left(x^{2} + y^{2} + z^{2}\right)^{\frac{3}{2}}}\\- \frac{y}{\left(x^{2} + y^{2} + z^{2}\right)^{\frac{3}{2}}}\\- \frac{z}{\left(x^{2} + y^{2} + z^{2}\right)^{\frac{3}{2}}}\end{matrix}\right]