Geometric interpretation of the derivative in 3 dimensional space

Question

In 2d derivative shows the slope of the tangent.

For instance, we have x' = x + y - z , y' = 5x + 10y - z, z' = -x - y - z. What is geometric interpretation of these derivatives?

Answer 1

If you plot the graph of a function $f(x,y)$ in 3D space, then $\dfrac{\partial f}{\partial x}(x_0,y_0)$ represents the slope of the tangent of the curve $x\mapsto f(x,y_0)$ at the point $x_0$.