Fractional-order neural networks have demonstrated promising capabilities for modeling long-term temporal dependencies, yet the fractional-order parameter is typically selected manually and remains fixed throughout training. This limitation prevents the network from adapting its activation characteristics to the analyzed data. In this work, we propose an end-to-end learning framework for fractional-order activations in Fractional Derivative Long Short-Term Memory (FD-LSTM) networks, where the fractional-order parameter is optimized jointly with the network weights using gradient-based optimization. Three parameterization strategies are investigated, employing one shared parameter, two activation-specific parameters, and four gate-specific parameters. The proposed approach was evaluated on a real industrial dataset acquired during optical fiber cable manufacturing for anomaly detection. Experimental results show that all FD-LSTM variants achieved higher aggregated performance metrics than the conventional LSTM and GRU models. The best-performing configuration achieved an MCC of 0.9963 compared with 0.9917 for the conventional LSTM baseline. Given the near-saturated baseline performance, the observed differences are interpreted cautiously and primarily as descriptive evidence of the potential benefit of learnable fractional-order activations. Beyond the quantitative improvement, the experiments reveal a consistent optimization pattern across all configurations. The input, forget, and output gates naturally converge to the classical activation (ν=1), whereas only the candidate gate benefits from fractional-order adaptation. Furthermore, nearly identical performance obtained by the 1ν, 2ν, and 4ν parameterizations indicates that increasing the number of learnable fractional-order parameters provides only marginal benefits. These findings contribute to a better understanding of fractional-order recurrent neural networks and demonstrate that adaptive fractional-order activations constitute an effective extension of conventional LSTM architectures for industrial time-series analysis.