## History and applications

### Applications

#### The derivative of \(x^n\)

The binomial theorem can be used to give a proof of the derivative of \(x^n\).

The derivative of \(x^n\) with respect to \(x\) is given by

\[ \dfrac{d(x^n)}{dx} = \lim_{h \to 0} \dfrac{(x+h)^n-x^n}{h}. \]Using the binomial theorem,

\begin{align*} \dfrac{(x+h)^n-x^n}{h}&= \dfrac{1}{h} \, \Bigg(\sum_{k=0}^{n}\dbinom{n}{k}x^{n-k}h^k-x^n\Bigg)\\ &=\dfrac{1}{h} \, \Bigg( \sum_{k=1}^{n}\dbinom{n}{k}x^{n-k}h^k\Bigg)\\ &=\sum_{k=1}^{n}\dbinom{n}{k}x^{n-k}h^{k-1}. \end{align*}Thus

\[ \lim_{h \to 0} \dfrac{(x+h)^n-x^n}{h}=nx^{n-1}. \]#### Approximations

The binomial theorem can be used to find approximations. For example for \(1.01^5\) we have

\[ (1+0.01)^5 \approx 1 + 5\times0.01+10\times0.01^2 = 1+ 0.05 + 10\times0.0001 = 1.051. \]Of course, with calculators, it is not necessary to go through such a process. However we can see that \((1+x)^n>1+nx\) for \(x>0\), which is a very useful inequality.