Description
With t=tanh(x/temperature), return -x*(1-t²)/temperature², the partial derivative with respect to temperature; return zero directly when 1-t² is numerically zero.
Hyperbolic tangent maps a real argument to values between -1 and 1. Input scaling or temperature changes its slope and location in input coordinates. A derivative with respect to temperature differs from differentiation with respect to the scalar input.
With t=tanh(x/temperature), return -x*(1-t²)/temperature², the partial derivative with respect to temperature; return zero directly when 1-t² is numerically zero.
When to use this tool
Use Tanh Temperature Sensitivity when checking the stated hyperbolic-tangent coordinate or derivative convention.
How to use the tool and how it works
- Enter x, temperature. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read derivative. Use the output convention below.
Require temperature>0. With t=tanh(x/temperature), return -x*(1-t²)/temperature², the partial derivative with respect to temperature; return zero directly when 1-t² is numerically zero.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| x | Finite scalar input coordinate; use the mathematical scale specified above. | Required. Finite number. |
| temperature | Positive dimensionless logit/input temperature multiplier under the stated convention. | Required. Finite number; strictly above 0. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| derivative | Partial derivative of tanh(x/temperature) with respect to temperature at fixed x. | Required. Finite number. |
{
"input": {
"x": 2,
"temperature": 2
},
"output": {
"derivative": -0.20998717080701307
}
}Limitations and assumptions
- Input x is held fixed; floating-point saturation can produce an exact zero despite a nonzero mathematical tail derivative.
- Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
- Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Numeric signed-zero results are exported as zero.
Alternative or Complementary methods
Use the complete explicit function and the intended parameter, branch and numerical conventions for symbolic, tensor or model-level differentiation; assess training behavior independently.
References
Similar or alternative tools
- Activation: Tanh Scalar Second Derivative
Let t=tanh(x).
- Tanh Inflection Distance
Return inflectionX=-bias/slope, signedDistance=x-inflectionX and argumentDistance=abs(slope)*abs(signedDistance).