Well the expression for the $\frac {d^n}{dt^n} p(t) $ is $V_n(t) p(t)$, where $V_n(t)$ is a polynomial of degree $n$, and $V_n(t)$ may be determined recursivelyy

$$ V_{n+1}(t)e^{-t^2/(2\tau^2)} = \frac{d}{dt}\left(V_{n}(t)e^{-t^2/(2\tau^2)}\right) $$

Solving for $V_{n+2}(t)$ we have

$$ V_{n+1} = \frac{d}{dt} V_{n}(t) - \frac{t}{2\tau^2} V_n(t)$$

These are the Hermite polynomials, they have some asymptotic expansions in terms of cossines.

The equation you gave is not correct, maybe an approximation, or some different parameterization.

Numerically evaluating the derivaties we see some resemblance of the original function.

function hermite;
  tau=pi;
  t = tau * linspace(-5, 5, 1000);
  for order = 4:2:20
    subplot(3,3,order/2-1)
    a = -1/(2*tau^2);
    P = V_coefs(order, a);
    plot(t, polyval(P(end:-1:1), t).*exp(a*t.^2));
    title(num2str(order, 'n=%d'))
  end
  
end

function coefs = V_coefs(order, a)
  coefs = 1;
  for ii = 1:order
    coefs = [(1:(ii-1)).*coefs(2:end), 0, 0] + [0, coefs .* (2 * a)];
  end
end

Well the expression for the $\frac {d^n}{dt^n} p(t) $ is $V_n(t) p(t)$, where $V_n(t)$ is a polynomial of degree $n$, and $V_n(t)$ may be determined recursivelyy

$$ V_{n+1}(t)e^{-t^2/(2\tau^2)} = \frac{d}{dt}\left(V_{n}(t)e^{-t^2/(2\tau^2)}\right) $$

Solving for $V_{n+2}(t)$ we have

$$ V_{n+1} = \frac{d}{dt} V_{n}(t) - \frac{t}{2\tau^2} V_n(t)$$

In order to link the two forms of the equation we need

$$\begin{eqnarray} g(t) &=& p(t)cos(2\pi f_{c}t) \\ &=& V_{TX}exp(-\frac{t^{2}}{2\tau^2})cos(2\pi f_{c}t) \\ &=& \frac{d^{n}}{dt^{n}} \left( V_{TX }exp(-\frac{t^{2}}{2\tau^2}) \right) \\ &=& p(t)V_n(t) \end{eqnarray}$$

Thus $cos(2\pi f_c t) = V_n(t)$, the right hand side is a polynomial. As $n \to \infty$, the are the Hermite polynomials have some asymptotic expansions in terms of cossines. So the two may assume approximate values for certain values of $f_c$ and $\tau$.

Numerically evaluating the derivaties we see some resemblance with gausiand windowed cosine.

function hermite;
  tau=pi;
  t = tau * linspace(-5, 5, 1000);
  for order = 4:2:20
    subplot(3,3,order/2-1)
    a = -1/(2*tau^2);
    P = V_coefs(order, a);
    plot(t, polyval(P(end:-1:1), t).*exp(a*t.^2));
    title(num2str(order, 'n=%d'))
  end
  
end

function coefs = V_coefs(order, a)
  coefs = 1;
  for ii = 1:order
    coefs = [(1:(ii-1)).*coefs(2:end), 0, 0] + [0, coefs .* (2 * a)];
  end
end