How to interpret elasticity of standardized (Z-Score) predictor in NBM

I have run a negative binomial regression for which I am calculating elasticity of the predictors using means (dy/dx)*(x/y). One of the predictors, dwell time, is unevenly distributed, so I am using a Z-Score of dwell time in the model. This predictor has a negative effect (Beta) on the outcome (as expected).
However, when calculating elasticity, the value turns positive because the "mean" of the standardized predictor is negative. In effect, I have a negative effect (Beta) on the outcome but a positive elasticity due to the mean of the standardized predictor being negative. Logically, that should mean a 1% increase in the predictor leads to a (+)% increase in the outcome.
How should i Interpret the elasticity? I ask because a negative (standardized) mean is distorting the direction of the expected effect.
This raises a broader question: If the mean of the Z-Score is zero, the above formula renders the elasticity zero. In that event, how should one compute elasticity when a predictor's Z-Score is being used in a model?
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