Develop a fit to the data $($given in the table$)$ of the form C $=A+Bt+D{t}^2+E{t}^3+F{t}^4$ where $A,B,D,E,$ and $F$ are constants. If the same fit to the data applies for $t$ up to $12,$ determine the value of $\frac{dC}{dt}($i$.$e$.,$ the first derivative of $C)$ at $t=5\mathrm{.}1.$

Express your answer to two decimal places. Be sure to carefully input and check your answer BEFORE submitting since there may be several digits. $($Note that in the future you may have more than $5$ data points, thereby making the use of a calculator somewhat tedious. Thus, you are encouraged to do this problem using your favorite math solving program that is capable of doing non$-$linear fits.$)$

$\backslash$table$[[t,1,2,3,4,5],[C,24\mathrm{.}5,192\mathrm{.}1,802\mathrm{.}3,2332\mathrm{.}1,5438\mathrm{.}5]]$