Formal Derivative of Polynomials Satisfies Leibniz's Rule

Theorem
Let $R$ be a commutative ring with unity.

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

Let $f, g \in R \left[{X}\right]$ be polynomials.

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

Then:
 * $\left({f g}\right)' = f g' + f' g$

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

Then:
 * $\left({X^n X^m}\right)' = \left({n + m}\right) X^{n + m}$

and:
 * $\left({X^n}\right)' X^m + X^n \left({X^m}\right)' = n X^n X^m + m X^n X^m$