The planar Chain Rule and the Differential Equation for the planar Logarithms

A planar monomial is by definition an isomorphism class of a finite, planar, reduced rooted tree. If x denotes the tree with a single vertex, any planar monomial is a non-associative product in x relative to m−array grafting. A planar power series f(x) over a field K in x is an infinite sum of K−multiples of planar monomials including the unit monomial represented by the empty tree. For every planar power series f(x) there is a universal differential df(x) which is a planar power series in x and a planar polynomial in a variable y which is the differential dx of x. We state a planar chain rule and apply it to prove that the derivative d dx (Expk(x)) is the k−ary planar exponential series. A special case of the planar chain rule is proved and it derived that the planar universe series Logk(1 + x) of Expk(x) satisfies the differential equation