แก้ไขปัญหาคณิตศาสตร์

ก้าวต่อก้าว

1. f(x, y) = \log{\left(x^{2} + y^{2} \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} = \frac{2 x}{x^{2} + y^{2}}

   Treat every variable except x as a constant.

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

   Treat every variable except y as a constant.

4. \nabla f = \left[\begin{matrix}\frac{2 x}{x^{2} + y^{2}}\\\frac{2 y}{x^{2} + y^{2}}\end{matrix}\right]

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