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Table 2.8.1 contains a graph of the principal branch for each of the six trig functions, and the graph of the corresponding inverse function.
Principal Branch

Principal Domain

Inverse Function


$\frac{\mathrm{\π}}{2}\le x\le \frac{\mathrm{\π}}{2}$



$0\le x\le \mathrm{\π}$



$\frac{\mathrm{\pi}}{2}\le x\le \frac{\mathrm{\pi}}{2}$



$0\le x\le \mathrm{\π}$



$0\le x\le \mathrm{\π}$



$\frac{\mathrm{\pi}}{2}\le x\le \frac{\mathrm{\pi}}{2}$


Table 2.8.1 Principal branches of the trig functions, and their inverse functions



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Table 2.8.2 lists the derivatives of the six inverse trigonometric functions.
Function

Maple's Derivative

Textbook Derivative

arcsine

$\frac{\ⅆ}{\ⅆx}\mathrm{arcsin}\left(x\right)$ = $\frac{{1}}{\sqrt{{1}{}{{x}}^{{2}}}}$${}$

$\frac{1}{\sqrt{1{x}^{2}}}$, $1<x<1$

arccosine

$\frac{\ⅆ}{\ⅆx}\mathrm{arccos}\left(x\right)$ = ${}\frac{{1}}{\sqrt{{1}{}{{x}}^{{2}}}}$${}$

$\frac{1}{\sqrt{1{x}^{2}}}$, $1<x<1$

arctangent

$\frac{\ⅆ}{\ⅆx}\mathrm{arctan}\left(x\right)$ = $\frac{{1}}{{1}{\+}{{x}}^{{2}}}$${}$

$\frac{1}{1\+{x}^{2}}$

arccotangent

$\frac{\ⅆ}{\ⅆx}\mathrm{arccot}\left(x\right)$ = ${}\frac{{1}}{{1}{\+}{{x}}^{{2}}}$${}$

$\frac{1}{1\+{x}^{2}}$

arcsecant

$\frac{\ⅆ}{\ⅆx}\mathrm{arcsec}\left(x\right)$ = $\frac{{1}}{{{x}}^{{2}}{}\sqrt{{1}{}\frac{{1}}{{{x}}^{{2}}}}}$${}$

$\frac{1}{\leftx\right\sqrt{{x}^{2}1}}$, $\leftx\right\>1$

arccosecant

$\frac{\ⅆ}{\ⅆx}\mathrm{arccsc}\left(x\right)$ = ${}\frac{{1}}{{{x}}^{{2}}{}\sqrt{{1}{}\frac{{1}}{{{x}}^{{2}}}}}$${}$

$\frac{1}{\leftx\right\sqrt{{x}^{2}1}}$, $\leftx\right\>1$

Table 2.8.2 Derivatives of the six inverse trigonometric functions



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Maple's differentiation formulas are correct for $x$ complex, but in the typical calculus text, the formulas are stated for $x$ real. That is why the textbook formulas have restrictions and vary slightly from the Maple form.
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The second column in the table uses the formal name of each inverse function. However, in 2D math mode, Maple understands the usage ${\mathrm{sin}}^{1}\left(x\right)$ for $\mathrm{arcsin}\left(x\right)$, etc.
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