Differentiation of $\cos v$

Let $y = \cos v$. By item 29, §1.1, this may be written

$$y = \sin \left ( \frac{\pi}{2} - v \right ).$$

Differentiating by formula (XI),

$$\begin{array}{ll} \frac{dy}{dx} & = \cos \left ( \frac{\pi... ...frac{dv}{dx} \right )\\ & = -\sin x \frac{dv}{dx}. \end{array}$$

(Since $\cos \left ( \frac{\pi}{2} \right ) = \sin v$, by 29, §1.1.) Therefore,

$$\frac{d}{dx} (\cos v) = -\sin v \frac{dv}{dx},$$

(equation (XII) in §5.1 above).