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Multivariable-Calculus · Partial-Derivatives

Gradient of t·x^2y

t x^{2} y

Step by step

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

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

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

    Treat every variable except t as a constant.

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

    Treat every variable except x as a constant.

  4. \frac{\partial f}{\partial y} = t x^{2}

    Treat every variable except y as a constant.

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

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

Reveal the answer
\nabla f = \left[\begin{matrix}x^{2} y\\2 t x y\\t x^{2}\end{matrix}\right]