My textbook gives the formula for compound interest as:
$A\left( t\right) =P\left( 1+\dfrac {r}{n}\right) ^{nt}$
Where: P = The principal, r=the annual rate of interest, n= the frequency of compounding, t=Time in years and A is the total interest accrued over time.
It then goes onto show how if we compound £1 continuously at a rate of 100% for 1 year, for greater and greater values of $n$ we get:
$\left( 1+\dfrac {1}{n}\right) ^{n}\rightarrow e$
And then uses this to derive the formula for continuously compounded interest:
$A\left( t\right) =Pe^{rt}$
The book says it uses "a little calculus and the definition of e" to derive this, but how exactly does it do this?
Let's see. The limit claim is pretty widely discussed on MSE, so I'm assuming you're willing to believe that the limit does approach $e$. Once you have that, you can look at the formula $$ A_n(t) = P \left(1+ \frac{r}{n}\right)^{nt} $$ and do a little fiddling. Let $m = \dfrac{n}{r}$, so that $n = rm$. Then rewrite: $$ A_n(t) = P \left(1+ \frac{r}{n}\right)^{nt} = P \left(1 + \frac{1}{m}\right)^{mrt}= P \left(\left(1 + \frac{1}{m}\right)^m \right)^{rt} $$
Now as $n \to \infty$, the thing inside the large parentheses approaches $e$, so you get
$$ A(t) = Pe^{rt}. $$
As for the main limit, the usual approach is to say you want to find $$ L = \lim_{n \to \infty} \left(1 + \frac{1}{n} \right)^n $$ but instead, you compute \begin{align} \ln L &= \lim_{n \to \infty} \ln \left(1 + \frac{1}{n} \right)^n \\ &= \lim_{n \to \infty} n \ln \left(1 + \frac{1}{n} \right) \\ &= \lim_{n \to \infty} \frac{\ln \left(1 + \frac{1}{n} \right)}{1/n}, \end{align} which you can evaluate with L'hopital's rule (take derivative of top and bottom, since both go towards 0): \begin{align} \ln L &= \lim_{n \to \infty} \frac{\ln \left(1 + \frac{1}{n} \right)}{1/n} \\ &= \lim_{n \to \infty} \frac{\frac{1}{1 + \frac{1}{n}}\left(\frac{-1}{n^2}\right)} {\frac{-1}{n^2}}\\ &= \lim_{n \to \infty} \frac{1}{1 + \frac{1}{n}}\\ &= 1. \end{align} Since the natural log of your limit is $1$, the limit itself must be $e$.$$$$$$