Differentiation of $\cot v$

Let $y = \cot v$. By item 26, §1.1, this may be written $y = \frac{1}{\tan v}$. Differentiating by formula VII,

$$\begin{array}{ll} \frac{dy}{dx} & = - \frac{\frac{d}{dx}(\t... ...dv}{dx}}{\tan^2 v} \\ & = -\csc^2 v \frac{dv}{dx}. \end{array}$$

Therefore,

$$\frac{d}{dx}(\cot v) = -\csc^2 v \frac{dv}{dx}$$

(equation (XIII) in §5.1 above).