Description
Return (z*sigmoid(z)-softplus(z))/beta², the beta partial derivative of log(1+exp(beta*x))/beta at fixed x.
Beta-scaled softplus is log(1+exp(beta*x))/beta for positive beta. Its input slope is a sigmoid, while curvature and parameter sensitivities are different derivatives. Numerical libraries may switch to a linear tail; such a switch is distinct from the smooth mathematical function.
Return (z*sigmoid(z)-softplus(z))/beta², the beta partial derivative of log(1+exp(beta*x))/beta at fixed x.
When to use this tool
Use Softplus Beta Partial Derivative when checking the stated softplus curvature, parameter sensitivity or branch convention.
How to use the tool and how it works
- Enter x, beta. 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 beta>0 and finite z=beta*x. Return (z*sigmoid(z)-softplus(z))/beta², the beta partial derivative of log(1+exp(beta*x))/beta at fixed x.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| x | Finite scalar input coordinate; use the mathematical scale specified above. | Required. Finite number. |
| beta | Supplied scalar shape parameter under the explicitly stated function convention. | Required. Finite number; strictly above 0. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| derivative | Partial derivative with respect to beta at fixed input x, not sigmoid(beta*x). | Required. Finite number. |
{
"input": {
"x": 0,
"beta": 1
},
"output": {
"derivative": -0.6931471805599453
}
}Limitations and assumptions
- The softplus inside the numerator has unit scale; this parameter derivative differs from the first input derivative sigmoid(beta*x).
- 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: Softplus Scalar Second Derivative
With s=sigmoid(beta*x), return beta*s*(1-s), the second derivative with respect to input x.
- Softplus Scalar Third Derivative
For beta-scaled softplus log(1+exp(beta*x))/beta with beta>0, require beta*x finite.