df is the REDUCE operator to compute the derivative. In df(f,x),
the operator df is to compute
the partial derivative of f with respect to b. Note that the x dependence of f should be declared in adavance.
If it is not the case, then REDUCE assumes that f is independent of x and gives 0.
The application of nested df operators realizes the computation of multiple derivatives.
df(df(f,x),y)
computes
the second-order derivative $\displaystyle \frac{\partial^2 f}{\partial y\partial x}.$
depend x,v,t;
states that x is a function of v and t. This is to declare the implicit dependence of variables.
Because of this declaration,
df(x^2,t)
is not 0 but $\displaystyle 2x\frac{dx}{dt}.$ Here, df(x,t) is not explicitly computed but the evaluation is delayed.
If you understand what the following is, then I hope you to explain what it is below!
Note that df(x,t,2)=
$\displaystyle \frac{d^2x}{dt^2}$ is the second-order time derivative of x.
}