Formal Derivative of Polynomials Satisfies Leibniz's Rule

Theorem
Let $R$ be a ring.

Let $R \left[{X}\right]$ be the ring of polynomial forms over $R$.

Let $f,g\in R[x]$.

Let $f'$ and $g'$ denote their formal derivatives.

Then $(fg)'=fg'+f'g$.

Proof
Both sides are bilinear functions of $f$ and $g$, so it suffices to verify the equality in the case where $f(x)=x^n$ and $g(x)=x^m$.

Then $(x^nx^m)'=(n+m)x^{n+m}$

and $(x^n)'x^m+x^n(x^m)'=nx^nx^m+mx^nx^m$.