Chapter 12.3, Problem 6P

Verify Problem 1. Hints: One method is to use mathematical induction. Another method is to write $\frac{d}{dx}(uv)=D(uv)=\left(D_u+D_v\right)(uv)$, where $D_u$ acts only on $u$ and $D_v$ acts only on v, that is, $D_u(uv)$ means $v(du/dx)$, etc. Then $\frac{d^n}{d{x}^n}(uv)={\left(D_u+D_v\right)}^n(uv)$.

Expand ${\left(D_u+D_v\right)}^n$ by the binomial theorem and interpret the terms to get Leibniz' rule.