In this paper, we consider Leibniz algebras with derivations. A pair consisting of a Leibniz algebra and a distinguished derivation is called a LeibDer pair. We define a cohomology theory for LeibDer pair with coefficients in a representation. We study central extensions of a LeibDer pair. In the next, we generalize the formal deformation theory to LeibDer pairs in which we deform both the Leibniz bracket and the distinguished derivation. It is governed by the cohomology of LeibDer pair with coefficients in itself. Finally, we consider homotopy derivations on sh Leibniz algebras and 2-derivations on Leibniz 2-algebras. The category of 2-term sh Leibniz algebras with homotopy derivations is equivalent to the category of Leibniz 2-algebras with 2-derivations.

在本文中，我们考虑带导数的莱布尼兹代数。由Leibniz代数和杰出的导数组成的对称为LeibDer对。我们为LeibDer对定义了一个表示系数的同调理论。我们研究LeibDer对的中心扩展。接下来，我们将形式变形理论推广到LeibDer对，其中我们对Leibniz托槽和专有推导都进行变形。它是由LeibDer对本身与系数的同调性决定的。最后，我们考虑sh莱布尼兹代数的同伦派生和莱布尼兹2代数的2派生。具有同态派生的2项sh莱布尼兹代数的类别与具有2个派生的莱布尼兹2代数的类别相同。