Solved – Alternative ways for interpretation of odds

I am reading the second edition of Categorical Data Analysis by Alan Agresti, and somehow stuck in the following second paragraph:
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I don't quite understand why $betapi(hat{x})(1 – pi(hat{x}))$ will give the probability when $x = 26.3$, can anyone enlighten me? Thanks.

The answer is near the bottom of p166. It's using a linear approximation (what social scientists would call a 'marginal effect'). A small change $delta x$ in $x$ gives a change in probability of: $$deltapi approx frac{partial pi(x)}{partial x} delta x.$$ With $operatorname{logit}(pi(x)) = alpha + beta x$, it's straightforward to show that $ partial pi(x) / partial x = beta pi(x)(1-pi(x))$.

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