What is spatial derivative?
The fourth-order spatial derivative is calculated by a finite difference (FD) derived specifically for fourth derivatives and by application of stage-wise differentiation in which a FD for second derivatives is used twice. From: Traveling Wave Analysis of Partial Differential Equations, 2012.
What is the derivative of velocity with respect to position?
In Physics
Velocity is the derivative of position with respect to time: v(t)=ddt(x(t)).