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In vector calculus

The product rule extends to various product operations of vector functions on R n {\displaystyle \mathbb {R} ^{n}}:

There are also analogues for other analogs of the derivative: if f and g are scalar fields then there is a product rule with the gradient:

Such a rule will hold for any continuous bilinear product operation. Let B : X × Y → Z be a continuous bilinear map between vector spaces, and let f and g be differentiable functions into X and Y, respectively. The only properties of multiplication used in the proof using the limit definition of derivative is that multiplication is continuous and bilinear. So for any continuous bilinear operation,

This is also a special case of the product rule for bilinear maps in Banach space.