Numbers and Sets | Part IA, 2002

If f,g:RRf, g: \mathbb{R} \rightarrow \mathbb{R} are infinitely differentiable, Leibniz's rule states that, if n1n \geqslant 1,

dndxn(f(x)g(x))=r=0n(nr)f(nr)(x)g(r)(x)\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.)

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