Description
Return -abs(x)-2*log1p(exp(-abs(x))), the natural logarithm of sigmoid's first derivative.
The logistic sigmoid is 1/(1+exp(-x)) and its inverse logit is log(p/(1-p)) on 0<p<1. Temperature and input sensitivities use different derivatives. The logarithm of a derivative is distinct from the derivative of a logarithm, especially when numerical tails underflow.
Return -abs(x)-2*log1p(exp(-abs(x))), the natural logarithm of sigmoid's first derivative.
When to use this tool
Use Sigmoid Log Derivative when checking the stated sigmoid/logit scalar derivative or stable product convention.
How to use the tool and how it works
- Enter x. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read logDerivative. Use the output convention below.
Return -abs(x)-2*log1p(exp(-abs(x))), the natural logarithm of sigmoid's first derivative.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| x | Finite scalar input coordinate; use the mathematical scale specified above. | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| logDerivative | Natural log of the sigmoid first derivative; not the derivative of log sigmoid. | Required. Finite number. |
{
"input": {
"x": 0
},
"output": {
"logDerivative": -1.3862943611198906
}
}Limitations and assumptions
- This is log(sigmoid′(x)), not the derivative of log(sigmoid(x)); it avoids forming an underflowing derivative before taking its logarithm.
- 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
- Sigmoid Scalar Third Derivative
With s=sigmoid(x), return s*(1-s)*(1-6*s+6*s²), the third scalar input derivative.
- Sigmoid Logit Jacobian Sign
Return sigmoidDerivative=p*(1-p), inverse-logit derivative=1/(p*(1-p)) and sign=1.