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Let $\theta = \tan^{-1}(t)$. Nowadays it is taught: 1º that $$ \frac{d\theta}{dt} = \frac 1{dt\,/\,d\theta} = \frac 1{1+t^2}, \tag1 $$ 2º that, via the fundamental theorem of calculus, this is ...

I'm not teaching calculus right now, but I talk to someone who does, and the question that came up is why emphasize the $h \to 0$ definition of a derivative to calculus students? Something a teacher ...