History of $\frac d{dt}\tan^{-1}(t)=\frac 1{1+t^2}$

I now believe that my question (and suggestion that proof $(1)$ should have become standard before Lacroix) relied on the misconception that tangent was easier to differentiate than arctangent. In fact the calculus of inverse trigonometric functions took off earlier, as has been explained by G. Eneström (1905), C. Boyer (1947), or V. J. Katz (1987):

it was quite common [at first] to deal with what we call the arcsine function rather than the sine.

C. Wilson (2001, 2007) concurs and stresses that our trig functions with periodic graphs weren’t much seen or differentiated until Euler “found” them to solve 2nd order linear differential equations (1741); so much so that he still wrote in (1749, p.15):

as this way of operating is not yet commonplace, it will be apropos to warn that the differentials of the formulas $\sin.$ : $\cos.$ : $\mathrm{tang}.$ : $\cot.$ are $d\,\cos.$ : $-d\,\sin.$ : $\smash[b]{\frac{d}{\cos.\,^2}}$ & $\smash[b]{-\frac{d}{\sin.\,^2}}$

— and e.g. in (1796, p.163) L’Huilier still computed $\tan'$ from $\arctan'$ rather than vice versa.

In this vein, $d = \frac{dt}{1+t^2}$ was not proved like $(1)$, but by a differential triangle argument similar to $(*)$ but simpler and attributed to Cotes (Aestimatio errorum, 1722): in modern notation, parametrize the unit circle with $(x,y)=\frac{(1,\,t)}{\sqrt{1+t^2}}$ and obtain $$ d\theta =\sqrt{\smash{dx^2+dy^2}\vphantom{a^2}} =\frac{dt}{1+t^2} \tag6 $$ ($=\mathrm{CE}$ in Cotes’ figure, which became standard in many books even before his own — the list could almost be described as “everyone but Euler”):

1708 Charles-René Reyneau §590 fig. 41   1718 John Craig pp.52–54   1722 Roger Cotes (posthumous) Lemma II   1730 Edmund Stone p.63 fig. 13   1736 James Hodgson p.230   1736 John Muller §247 fig. 153  1737 Thomas Simpson §143   1742 Colin MacLaurin §195 fig. 52   1743 William Emerson pp.171–172 fig. 76   1748 Maria Agnesi p.639 fig. 4   1749 Charles Walmesley (credits Cotes) pp.3,53 fig. 10   1749 William Emerson p.29 fig. 6   1750 Thomas Simpson §142   1754 Louis-Antoine de Bougainville p.24 fig. 9   1761 Abraham Kästner  §299 fig. 18   1765 Jean Le Rond D’Alembert p.640 fig. 25   1767 Étienne Bézout p.146 fig. 46   1768 Thomas Le Seur & François Jacquier p.63 fig. 7   1774 Jean Saury pp.25,63 fig. 3   1779 Samuel Horsley pp.298–299   1786 Simon L’Huilier pp.103–104 fig. 20   1795 Simon L’Huilier §76 fig. 17 

As to our usual proof $(1)$, it appears before Lacroix in 18-year-old Legendre’s Theses mathematicæ (1770), then in a book by their common teacher J.-F. Marie and several others:

1770 Adrien-Marie Legendre pp.10,16   1772 Joseph-François Marie §904   1777 Jacques-Antoine Joseph Cousin p.81   1781 Claude Bertrand p.140   1781 Louis Lefèvre-Gineau p.31   1795 Simon L’Huilier §76   1797 Sylvestre-François Lacroix pp.113–114   1801 Joseph-Louis Lagrange p.81   1810 Sylvestre-François Lacroix pp.lii,203–204  

The Madhava–Gregory series, by R. C. Gupta, attributes (3) to Indian mathematician-astronomer Madhava of Sangamagrama (circa 1350–1425). He also writes that a geometric derivation which is basically equivalent to (1) can be found in the book Yuktibhāṣā written by Indian astronomer Jyesthadeva (circa 1500-1601) of the Kerala school of mathematics in about 1530.