tanh2x
tanh2x is the hyperbolic tangent function evaluated at the argument 2x, i.e., tanh(2x). It is a scaled variant of the basic tanh function and is commonly used in mathematics and applications as a smooth, S-shaped response.
- tanh(2x) = (e^{2x} − e^{−2x})/(e^{2x} + e^{−2x}) = (e^{4x} − 1)/(e^{4x} + 1).
- It satisfies the double-angle identity tanh(2x) = 2 tanh(x) / (1 + tanh^2(x)).
- It is related to the logistic form via tanh(x) = 2/(1 + e^{−2x}) − 1.
- It is an odd function: tanh(−2x) = −tanh(2x).
- It is strictly increasing with range (−1, 1); limits at infinity are 1 and −1 respectively.
- The derivative is d/dx tanh(2x) = 2 sech^2(2x), where sech(u) = 1/cosh(u). The second derivative is −8 sech^2(2x)
- The indefinite integral ∫ tanh(2x) dx = (1/2) ln(cosh(2x)) + C.
- Its Taylor series around x = 0 begins as tanh(2x) = 2x − (8/3)x^3 + (64/15)x^5 − ...
- As an activation function in neural networks, tanh(2x) offers outputs in (−1, 1) and a steeper
- It also appears in modeling smooth, sigmoidal responses in physics and engineering.
- When evaluating, ensure the argument is indeed 2x; otherwise tanh(x) and tanh(2x) have different properties and