Implement Dense Layer with NumPy

Problem

Implement a fully connected neural network layer using only NumPy. Given an input matrix x of shape (batch_size, in_features), a weight matrix W of shape (in_features, out_features), and a bias vector b of shape (out_features,), write a function that returns the forward output y = x @ W + b and the gradients dx, dW, and db for a provided upstream gradient dout of shape (batch_size, out_features). The implementation must be fully vectorized and rely on NumPy broadcasting rather than Python loops.

Constraints

• 1 <= batch_size <= 10^4
• 1 <= in_features, out_features <= 10^3
• x.shape == (batch_size, in_features)
• W.shape == (in_features, out_features)
• b.shape == (out_features,)
• dout.shape == (batch_size, out_features)
• Use NumPy operations only; avoid explicit Python loops over samples or features