Activation: Softplus Scalar Second Derivative Tool

Description

With s=sigmoid(beta*x), return beta*s*(1-s), the second derivative with respect to input 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.

With s=sigmoid(beta*x), return beta*s*(1-s), the second derivative with respect to input x.

When to use this tool

Use Activation: Softplus Scalar Second 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 result. Expand result to inspect the returned fields.

Require beta>0 and finite beta*x. With s=sigmoid(beta*x), return beta*s*(1-s), the second derivative with respect to input 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.

Output field	Interpretation	Returned shape
result	Object containing the operation-specific returned members listed below.	Required. Object.
result.secondDerivative	Second Derivative	Calculated result member: number.

json

{
  "input": {
    "x": 0,
    "beta": 2
  },
  "output": {
    "result": {
      "secondDerivative": 0.5
    }
  }
}

Limitations and assumptions

Numerical saturation can yield zero curvature; this does not select a beta or predict model-training behavior.
Outputs are nested inside result. 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

Calculation and input contract: activation/softplus-scalar-second-derivative — tool.io

Similar or alternative tools

Softplus Scalar Third Derivative
For beta-scaled softplus log(1+exp(beta*x))/beta with beta>0, require beta*x finite.
Softplus Beta Partial Derivative
Return (z*sigmoid(z)-softplus(z))/beta², the beta partial derivative of log(1+exp(beta*x))/beta at fixed x.