of a product is obtained by multiplying both members of this equation by the product of the Higher derivatives It can also be generalized to the general Leibniz rule for the nth derivative of a product of two factors, by symbolically expanding according to the binomial theorem: Applied at a specific point x, the above formula gives: Furthermore, for the nth derivative of an arbitrary number of factors, one has a similar formula with multinomial coefficients: Higher partial derivatives For partial derivatives, we have where the index runs through all subsets of , and is the cardinality of . For example, when , Banach space Suppose X, Y, and Z are Banach spaces (which includes Euclidean space) and B : X × Y → Z is a continuous bilinear operator. Then B is differentiable, and its derivative at the point (x,y) in X × Y is the linear map D(x,y)B : X × Y → Z given by This result can be extended to more general topological vector spaces. In vector calculus The product rule extends to various product operations of vector functions on : For scalar multiplication: For dot product: For cross product of vector functions on : 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. Derivations in abstract algebra and differential geometry In abstract algebra, the product rule is the defining property of a derivation. In this terminology, the product rule states that the derivative operator is a derivation on functions. In differential geometry, a tangent vector to a manifold M at a point p may be defined abstractly as an operator on real-valued functions which behaves like a directional derivative at p: that is, a linear functional v which is a derivation, Generalizing (and dualizing) the formulas of vector calculus to an n-dimensional manifold M,