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We derive bounds on the error, in high-order Sobolev norms, incurred in the approximation of Sobolev-regular as well as analytic functions by neural networks with the hyperbolic tangent activation function. These bounds provide explicit estimates on the approximation error with respect to the size of the neural networks. We show that tanh neural networks with only two hidden layers suffice to approximate functions at comparable or better rates than much deeper ReLU neural networks. • Explicit bounds for function approximation in Sobolev norms by tanh neural networks. • Tanh networks with 2 hidden layers are at least as expressive as deeper ReLU networks. • Improved convergence rate for neural network approximation of analytic functions. [ABSTRACT FROM AUTHOR]