# Checking if first derivative is zero at a given point

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.

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@rcollyer beat my by 5 seconds :) –  tkott Mar 20 '12 at 19:16
@tkott, I win again. :P –  rcollyer Mar 20 '12 at 19:17
@Martin are you sure you want Dt instead of D? If you want Dt, then you will need to specify all combinations of Dt[h, tau] through rules –  tkott Mar 20 '12 at 19:18

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]

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You don't have to calculate the derivatives one by one. To calculate all the first order partial derivatives at the specified point in one go you can do something like D[V[a, h, tau], {{a, h, tau}}] /. {a -> 8.47344, h -> 1.67718, tau -> 12.9438} –  Heike Mar 20 '12 at 22:09