Differentiation of ${\rm arccsc}\, v$

Let

$$y =\ {\rm arccsc}\, v;$$

then

$$v =\ \csc\, y.$$

This function is defined for all values of $v$ except those lying between $-1$ and $+1$, and is seen to be many-valued. To make the function single-valued, $y$ is taken as the arc of smallest numerical value whose cosecant is $v$. This means that if $v$ is positive, we confine ourselves to points on the arc $AB$ (Figure 5.11), $y$ taking on values between 0 and $\frac{\pi}{2}$ ( $\frac{\pi}{2}$ may be included); and if $v$ is negative, we confine ourselves to points on the arc $CD$, $y$ taking on values between $-\pi$ and $-\frac{\pi}{2}$ ( $-\frac{\pi}{2}$ may be included).

Figure 5.11: The inverse secant function ${\rm arccsc}\x$ using SAGE.

Figure 5.12: The standard branch of ${\rm arccsc}\x$ using SAGE.

Differentiating with respect to $y$ by XVI and following the method of the last section, we get

$$\frac{d}{dx}({\rm arccsc}v) = -\frac{\frac{dv}{dx}}{v\sqrt{v^2 - 1}}$$

(equation (XXIII) in §5.1 above).