Find the minimum speed of a particle with trajectory $\displaystyle c(t) = (t^3, t^{-2}), t \geq \frac{1}{2} $

So here's what I told my professor and he confirmed it as right (though I am still unsure if it was really right):

1. Write the speed formula:

$\displaystyle speed = \sqrt{ 9t^4 + 4t^{-6} }$

2. Write the derivative of the speed and find where it's equal to 0:

$\displaystyle \frac{d}{dt}( \sqrt{ 9t^4 + 4t^{-6} } )$

3. Write the second derivative of the speed and find where it's > 0:

$\displaystyle \frac{d}{dt}{(\frac{d}{dt}( \sqrt{ 9t^4 + 4t^{-6} } ))}$

So if this is correct, is there a way to simplify this so that I won't need to expand the equation that much? Or am I doing something wrong?