If you open the Negative Exponential.jmp nonlinear sample data example, the actual formula for the Nonlinear column looks something like this:
Parameter({b0=0.5, b1=0.5}, b0*(1-Exp(-b1*X)))
b0 * First(T#1=1-(T#2=Exp(-b1*X)), T#3=-(-1*T#2*X))
The Derivative of Model with respect to the parameters is:
{T#1, T#3*b0}
When the prediction model needs additional subexpressions evaluated, it uses the First function, which returns the value of the first argument expression, and also evaluates the other arguments. In this case additional assignments are needed for derivatives.
The derivative table itself is a list of expressions, one expression for each parameter to be fit. For example, the derivative of the model with respect to b0 is T#1; its thread in the prediction model is 1–(Exp(-b1*X)). The derivative with respect to b1 is T#3*b0, which is –(–1*Exp(-b1*X)*X)*b0 if you substitute in the assignments above. Although many optimizations are made, it does not always combine the operations optimally. You can see this by the expression for T#3, which does not remove a double negation.
If the derivative mechanism does not know how to take the analytic derivative of a function, then it takes numerical derivatives, using the NumDeriv function. If this occurs, the platform shows the delta that it used to evaluate the change in the function with respect to a delta change in the arguments. You might need to experiment with different delta settings to obtain good numerical derivatives.
If(Y==0, Log(1/(1+Exp(model))), Log(1 - 1/(1 + Exp(model))));
If(Y==0, Log(temp), Log(1-temp));

Help created on 9/19/2017