<Dl> <Dd> ∇ f = ∂ f ∂ x = (∂ f ∂ x 1 ∂ f ∂ x 2 ∂ f ∂ x 3) T . (\ displaystyle \ nabla f = (\ frac (\ partial f) (\ partial \ mathbf (x))) = (\ begin (bmatrix) (\ frac (\ partial f) (\ partial x_ (1))) & (\ frac (\ partial f) (\ partial x_ (2))) & (\ frac (\ partial f) (\ partial x_ (3))) \ \ \ end (bmatrix)) ^ (T).) </Dd> </Dl> <Dd> ∇ f = ∂ f ∂ x = (∂ f ∂ x 1 ∂ f ∂ x 2 ∂ f ∂ x 3) T . (\ displaystyle \ nabla f = (\ frac (\ partial f) (\ partial \ mathbf (x))) = (\ begin (bmatrix) (\ frac (\ partial f) (\ partial x_ (1))) & (\ frac (\ partial f) (\ partial x_ (2))) & (\ frac (\ partial f) (\ partial x_ (3))) \ \ \ end (bmatrix)) ^ (T).) </Dd> <P> More complicated examples include the derivative of a scalar function with respect to a matrix, known as the gradient matrix, which collects the derivative with respect to each matrix element in the corresponding position in the resulting matrix . In that case the scalar must be a function of each of the independent variables in the matrix . As another example, if we have an n - vector of dependent variables, or functions, of m independent variables we might consider the derivative of the dependent vector with respect to the independent vector . The result could be collected in an m × n matrix consisting of all of the possible derivative combinations . There are, of course, a total of nine possibilities using scalars, vectors, and matrices . Notice that as we consider higher numbers of components in each of the independent and dependent variables we can be left with a very large number of possibilities . </P> <P> The six kinds of derivatives that can be most neatly organized in matrix form are collected in the following table . </P>

Derivative of a vector with respect to a vector
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