Checking if first derivative is zero at a given point

Question

I have a function

V[a_, h_, tau_] := (Sqrt[3]/2) h (Sqrt[3] a + h/3 tau)^2

now I want to check, whether its first derivative is zero in a given point. Say for the solution

{a -> 8.47344, h -> 1.67718, tau -> 12.9438}

I tried

Dt[V[a, h, t], a, h, tau] /. {a -> 8.47344, h -> 1.67718, tau -> 12.9438} 

Is this the proper way to show that? I use V in a system of equations, which I am trying to solve with Newton's method.

Answer 1

You have three variables, so the "first derivative" is a little ambiguous. You can take a derivative with respect to one of these and feed the solution in directly, e.g.

D[V[a, h, tau], a] /. {a -> 8.47344, h -> 1.67718, tau -> 12.9438}

(Spoiler: none of them are zero)

When you take the total derivative Dt[V[a, h, t], a, h, tau], it gives you a result in terms of the relationship between the variables, such as Dt[a,h] or Dt[h,tau]. You have to specify what these are from you original problem. Substituting in numbers directly gives nonsensical expression such as Dt[8.47344, 1.67718]. You can't take the derivative of a number with respect to a number.

You could also ask Mathematica when one of the first derivatives is zero:

Solve[D[V[a, h, tau], a] == 0]