Calculus and Physics Help! If a particle's position is given by $x = 4-12t+3t^2$ (where $t$ is in seconds and $x$ is in meters):
a) What is the velocity at $t = 1$ s?
Ok, so I have an answer:
$v = \frac{dx}{dt} = -12 + 6t$
At $t = 1$, $v = -12 + 6(1) = -6$  m/s
But my problem is that I want to see the steps of using the formula $v = \frac{dx}{dt}$ in order to achieve $-12 + 6t$...
I am in physics with calc, and calc is only a co-requisite for this class, so I'm taking it while I'm taking physics. As you can see calc is a little behind. We're just now learning limits in calc, and I was hoping someone could help me figure this out.

You see the problem here is that the question is asking for a velocity at $t=1$. This means that they require and instantaneous velocity which is by definition the derivative of the position function at $t=1$. If you don't want to use derivative rules for some reason and you don't mind a little extra work then you can calculate the velocity from a limit. (In reality this is the same thing as taking the derivative as you will later see. A derivative is just a limit itself.)
To take the instantaneous velocity, we use the formula $\overline{v} = \frac{\Delta x}{\Delta t}$. I use $\overline{v}$ to denote the average velocity. Between time $t$ and $t+\Delta$ we have
$$\overline{v} = \frac{x(t+\Delta t) - x(t)}{\Delta t} = \frac{\left(3(t+\Delta t)^2 - 12(t+\Delta t)+4\right)-\left(3t^2-12t+4\right)}{\Delta t} $$
Simplifying a bit
$$\overline{v} = \frac{6t\Delta t + (\Delta t)^2 -12\Delta t}{\Delta t}$$
Now comes the calculus. If we take the limit as $\Delta t \rightarrow 0$, that is if we take the time interval to be smaller and smaller so that the average velocity approaches the instantaneous, then the instantaneous velocity $v$ is
$$v = \lim_{\Delta t\rightarrow 0}\frac{6t\Delta t + (\Delta t)^2 -12\Delta t}{\Delta t}=\lim_{\Delta t\rightarrow 0}\left(6t-12+\Delta t\right)=6t-12$$
Notice that this is exactly the derivative you had before and the above steps are in fact a calculation of the derivative from first principles.
