If $f, g: \mathbb{R} \rightarrow \mathbb{R}$ are infinitely differentiable, Leibniz's rule states that, if $n \geqslant 1$,

$$\frac{d^{n}}{d x^{n}}(f(x) g(x))=\sum_{r=0}^{n}\left(\begin{array}{c} n \\ r \end{array}\right) f^{(n-r)}(x) g^{(r)}(x)$$

Prove this result by induction. (You should prove any results on binomial coefficients that you need.)