Q:

How would I go about differentiating t with respect to x? 

$$t=\frac{x}{v}+\frac{\sqrt{(\frac{a}{2})^2+(a-x)^2}}{(\frac{v}{3})}$$
We haven't covered how to differentiate equations like this yet.  How would I go about finding dt/dx? 

A:

Here's a sneaky way around some of the work others have suggested.  In particular, you don't need to know how to differentiate the square-root function.  
Do a little rewriting via algebra:
\begin{align}
t&=\frac{x}{v}+\frac{\sqrt{(\frac{a}{2})^2+(a-x)^2}}{(\frac{v}{3})}\\
t-\frac{x}{v} &=\frac{\sqrt{(\frac{a}{2})^2+(a-x)^2}}{(\frac{v}{3})}\\
\frac{v}{3}\left(t-\frac{x}{v}\right) &=\sqrt{(\frac{a}{2})^2+(a-x)^2}\\
\left(\frac{v}{3}\right)^2\left(t-\frac{x}{v}\right)^2 &=(\frac{a}{2})^2+(a-x)^2
\end{align}
Now remembering that $t$ is really $t(x)$, a function that varies with $x$, we have a equality between two functions -- one on the left, one on the right.  So their derivatives must be equal.  The one on the left needs the power rule and the chain rule; the one on the right needs only the power rule.  Differentiating, we get
\begin{align}
2\left(\frac{v}{3}\right)^2\left(t-\frac{x}{v}\right)\left(\frac{dt}{dx} + \frac{-1}{v}\right) &=2(a-x)(-1)\\
\left(\frac{v}{3}\right)^2\left(t-\frac{x}{v}\right)\left(\frac{dt}{dx} + \frac{-1}{v}\right) &=(x-a)\\
\left(t-\frac{x}{v}\right)\left(\frac{dt}{dx} + \frac{-1}{v}\right) &=(x-a)\left(\frac{3}{v}\right)^2\\
\frac{dt}{dx} + \frac{-1}{v} &=\frac{(x-a)\left(\frac{3}{v}\right)^2}{t-\frac{x}{v}}\\
\frac{dt}{dx} &=\frac{(x-a)\left(\frac{3}{v}\right)^2}{t-\frac{x}{v}} + \frac{1}{v}\\
\end{align}
At this point, you have a formula for $\frac{dt}{dx}$, but it involves $t$, which may seem objectionable to you (or your teacher).  Fortunately, you started out with a formula that expresses $t$ in terms of $x$, and can plug that in to get 
\begin{align}
\frac{dt}{dx} &=\frac{(x-a)\left(\frac{3}{v}\right)^2}{t-\frac{x}{v}} + \frac{1}{v}\\ 
&=\frac{(x-a)\left(\frac{3}{v}\right)^2}{\left(\frac{x}{v}+\frac{\sqrt{(\frac{a}{2})^2+(a-x)^2}}{(\frac{v}{3})} \right)-\frac{x}{v}} + \frac{1}{v}\\
&=\frac{(x-a)\left(\frac{3}{v}\right)^2}{\left(\frac{\sqrt{(\frac{a}{2})^2+(a-x)^2}}{(\frac{v}{3})} \right)} + \frac{1}{v}\\
\end{align}
and there's your answer without using anything other than the chain rule and the (integer) power rule and a bit of algebra.  It can, of course, be simplified via a bit more algebra, but I leave that to you.