The development of programmable or trainable molecular circuits is an important goal in the field of molecular programming. Multilayer, nonlinear, artificial neural networks are a powerful framework for implementing such functionality in a molecular system, as they are provably universal function approximators. Here, we present a design for multilayer chemical neural networks with a nonlinear hyperbolic tangent transfer function. We use a weight perturbation algorithm to train the neural network which uses a simple construction to directly approximate the loss derivatives required for training. We demonstrate the training of this system to learn all 16 two-input binary functions from a common starting point. This work thus introduces new capabilities in the field of adaptive and trainable chemical reaction network (CRN) design. It also opens the door to potential future experimental implementations, including DNA strand displacement reactions.