  Machine learning techniques have recently gained prominence in physics,
yielding a host of new results and insights. One key concept is that of
backpropagation, which computes the exact gradient of any output of a program
with respect to any input. This is achieved efficiently within the
differentiable programming paradigm, which utilizes automatic differentiation
(AD) of each step of a computer program and the chain rule. A classic
application is in training neural networks. Here, we apply this methodology
instead to the numerical renormalization group (NRG), a powerful technique in
computational quantum many-body physics. We demonstrate how derivatives of NRG
outputs with respect to Hamiltonian parameters can be accurately and
efficiently obtained. Physical properties can be calculated using this
differentiable NRG scheme--for example, thermodynamic observables from
derivatives of the free energy. Susceptibilities can be computed by adding
source terms to the Hamiltonian, but still evaluated with AD at precisely zero
field. As an outlook, we briefly discuss the derivatives of dynamical
quantities and a possible route to the vertex.
