The Sum and Product Rule for Derivatives

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Problem 67 (The Sum and Product Rule for Derivatives)

Suppose that $F$ is a field and that $f(x),g(x)\in F[x]$.

1. Prove the sum rule, namely that $(f+g)'(x)=f'(x)+g'(x)$.
2. Prove the product rule, namely that $(fg)'(x)=f'(x)g(x)+f(x)g'(x)$.
3. What properties of the field $F$ were needed to complete your proofs? State a stronger theorem that requires less assumptions.

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