Sigmoid Function (Graph)
Calculate and graph the tables for the sigmoid function, as well as its first and second derivatives.
The sigmoid function is used in neural networks as an activation function, among other applications.
| \( \normalsize Sigmoid\ function\ \varsigma_{\alpha}(x)\\ \varsigma_{\alpha}(x) = \large \frac{1}{1+e^{-\alpha x}} = \large \frac {\tanh(\alpha x /2) + 1}{2}\\ \normalsize \varsigma '_{\alpha}(x) = \alpha \varsigma_{\alpha}(x) \{ 1 - \varsigma_{\alpha}(x) \} \\ \normalsize \varsigma ''_{\alpha}(x) = \alpha^{2} \varsigma_{\alpha}(x) \{ 1- \varsigma_{\alpha}(x) \} \{ 1 - 2 \varsigma_{\alpha}(x) \} \\ \) |
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