It is well-known that if $F$ is a differentiable vector field equal to the gradient of some potential function, then $F$ is curl free. Conversely, if $F$ is a curl free vector field on a simply connected domain, then $F$ is the derivative of a potential function.

In this talk, we present a free analog to this pair of results: the derivative of an analytic free map must be free-curl free and when we are on a connected free domain, every free-curl free map is a derivative of a free potential function.

Recall that a free function in $g$ freely noncommuting variables sends $g$-tuples of matrices (of the same size) to $h$-tuples of matrices (of the same size). Analytic free functions have remarkable properties. Most notably, the free derivative can be realized via point evaluation; a fact from which we profit greatly. Our proof follows the classical trajectory but with a strong free analytic flavor.