Differentiation of ${\rm arccot}u$

Let $y = {\rm arccot}\ v$; then $y = \cot\ y$. This function is defined for all values of $v$, and is many-valued.In order to make it single-valued, only values of $y$ between 0 and $\pi$ are considered; that is, the smallest positive arc whose cotangent is $v$.

Following the method of the last section, we get

$$\frac{d}{dx}({\rm arccot}\, v) = -\frac{\frac{dv}{dx}}{1 + v^2}$$

(equation (XXI) in §5.1 above).