Product rule for derivatives

For functions differentiable at $x$ , the product rule is

(fg)'(x)=f'(x)g(x)+f(x)g'(x).

To prove it, write the difference quotient as

\frac{f(x+h)-f(x)}h g(x+h) +f(x)\frac{g(x+h)-g(x)}h

and use continuity of the differentiable function $g$ .

Partial and bilinear versions

Holding the other variables fixed gives $\partial_j(fg)=(\partial_j f)g+f\partial_jg$ . More generally, for a fixed continuous bilinear map $B$ , $\partial_jB(u,v)=B(\partial_j u,v)+B(u,\partial_jv)$ . This includes dot products, cross products, outer products and matrix multiplication. The order of factors is retained when the product is noncommutative.