Editing Bell polynomials (section)

===Derivatives===

The partial derivatives of the complete Bell polynomials are given by{{Sfn|Bell|1934|loc=identity (5.1) on p. 266}}
: <math> \frac{\partial B_{n}}{\partial x_i} (x_1, \ldots, x_{n}) = \binom{n}{i} B_{n-i}(x_1, \ldots, x_{n-i}).</math>

Similarly, the partial derivatives of the partial Bell polynomials are given by
: <math> \frac{\partial B_{n,k}}{\partial x_i} (x_1, \ldots, x_{n-k+1}) = \binom{n}{i} B_{n-i,k-1}(x_1, \ldots, x_{n-i-k+2}).</math>

If the arguments of the Bell polynomials are one-dimensional functions, the chain rule can be used to obtain
: <math> \frac{d}{dx} \left(B_{n,k}(a_1(x), \cdots, a_{n-k+1}(x))\right) = \sum_{i=1}^{n-k+1} \binom{n}{i} a_i'(x) B_{n-i,k-1}(a_1(x), \cdots, a_{n-i-k+2}(x)).</math>