Derivative a interaction term in GAM Model - r

Question

I'm trying to model an estimation of the price elasticity of demand for each customer using GAM model, a model like this:
\ln D = \ln P + \ln P \cdot \sum_{i=1}^{20} f(X_i)
PED = \frac{\partial \ln D {\partial \ln P} = 1 + \sum_{i=1}^{20} f(X_i)
https://latex.codecogs.com/svg.image?$$&space;\ln&space;D&space;=\ln&space;P&space;+&space;\ln&space;P&space;\cdot&space;\sum_{i=1}^{20}&space;f(X_i)\\PED&space;=&space;\frac{\partial&space;\ln&space;D}{\partial&space;\ln&space;P}&space;=&space;1&space;+&space;&space;\sum_{i=1}^{20}&space;f(X_i)
where $D$ is Demand, $P$ is rate, PED is price elasticity of demand and $X_i$ is a set of customer's variable.
Since $PED$ is not observable, i want to estimate PED from the model created for log demand using gam model, but I have trying some difficulty in how to estimate that way.
I tried to get the each splines to calculate PED, but i failed. I know there is a package called gratia with derivatives function, but i dont understand how to use it to calculate ped.
Once the model to estimate demand is created, I will need to estimate the price elasticity of demand for each customer, but for these customers I don't have the rate variable, only the 20 personal variables.
I read some links:
https://stats.stackexchange.com/questions/495775/first-derivative-of-fitted-gam-changes-according-to-specified-model-distribution
https://stats.stackexchange.com/questions/590167/how-can-i-calculate-a-derivative-of-a-global-smooth-and-group-level-smooths-with
https://stats.stackexchange.com/questions/32013/what-is-the-mathematical-model-formula-corresponding-to-this-gam-model-fit-in-r
Really appreciate for any explanation, advices or other way to model my data.
Thanks
EDIT
What i've tried:
#create the dataset
A <- sample(x = 0:1000, size = 5000, replace = TRUE)
B <- sample(x = 0:1000, size = 5000, replace = TRUE)
C <- sample(x = 0:1000, size = 5000, replace = TRUE)
D <- sample(x = 0:1000, size = 5000, replace = TRUE)
log.R <- log(rbeta(5000, 5,10)*10) #log rate
log.Y <- log(rgamma(5000, 10, 20)*10000) #log demand
mydata <- data.frame(A, B, C, D, log.R, log.Y)
#the model
model <- gam(log.Y ~ s(A, by=log.R) + s(B, by=log.R) + s(C, by=log.R) + s(D, by=log.R), data = mydata, method = "REML")
mfx <- marginaleffects(model, variables = "log.R", eps = 10^-5)
head(mfx)
mfx returns a 'dydx' column, is it the elasticity of my data used to model?
And when i will apply this model to newdata, i got an error:
newdat = data.frame(A = 750, B = 500, C = 398, D = 740)
marginaleffects(model, variables = "log.R", eps = 10^-5, newdata= newdat, slope = 'dydx')
Error: There is no valid predictor variable. Please change the `variables` argument or supply a new data frame to the `newdata` argument.
What should I do?