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Multivariable-Calculus · Partial-Derivatives

Gradient of sin(x·y)

\sin{\left(x y \right)}

Langkah demi langkah

  1. f(x, y) = \sin{\left(x y \right)}

    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 \cos{\left(x y \right)}

    Treat every variable except x as a constant.

  3. \frac{\partial f}{\partial y} = x \cos{\left(x y \right)}

    Treat every variable except y as a constant.

  4. \nabla f = \left[\begin{matrix}y \cos{\left(x y \right)}\\x \cos{\left(x y \right)}\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.

Tunjukkan jawapan
\nabla f = \left[\begin{matrix}y \cos{\left(x y \right)}\\x \cos{\left(x y \right)}\end{matrix}\right]