Use principal inverse-trigonometric ranges

Inverse trigonometric functions use principal ranges

Read the function domain and principal range before differentiating. Then check the stricter conditions needed by the displayed derivative formula; endpoint composites require their own analysis.

Here $\sin^{-1}$, $\cos^{-1}$ and $\tan^{-1}$ mean arcsine, arccosine and arctangent. They are not $1/\sin$, $1/\cos$ or $1/\tan$; the source explicitly distinguishes inverse functions from reciprocals.

Inverse function: $\sin^{-1}x$

• Real domain: $[-1,1]$
• Principal range: $[-\pi/2,\pi/2]$

Inverse function: $\cos^{-1}x$

• Real domain: $[-1,1]$
• Principal range: $[0,\pi]$

Inverse function: $\tan^{-1}x$

• Real domain: $\mathbb R$
• Principal range: $(-\pi/2,\pi/2)$

For a differentiable inner function u, the three source basic/general rows are:

Function: $\sin^{-1}u$

• Basic derivative (u=x): $1/\sqrt{1-x^2}$
• General derivative: $u^{\prime}/\sqrt{1-u^2}$

Function: $\cos^{-1}u$

• Basic derivative (u=x): $-1/\sqrt{1-x^2}$
• General derivative: $-u^{\prime}/\sqrt{1-u^2}$

Function: $\tan^{-1}u$

• Basic derivative (u=x): $1/(1+x^2)$
• General derivative: $u^{\prime}/(1+u^2)$

The source shades the three basic formulas as being in MF27. It prints $-1\le x\le1$ (and $-1\le u\le1$) beside the first two derivative rules. Those are the function domains: the displayed derivative quotients require $|x|<1$ or $|u|<1$ so their denominators are nonzero.

Endpoint composites need separate analysis; a constant inner function can, for example, give a constant endpoint value with derivative zero.

The arctangent rule has no corresponding real endpoint restriction.

Why arcsine uses the positive square root

The source proof sets $y=\sin^{-1}x$, so $\sin y=x$.

Implicit differentiation gives $\cos y\,y^{\prime}=1$ and $y^{\prime}=1/\cos y$.

On the principal range $[-\pi/2,\pi/2]$, $\cos y\ge0$, hence $\cos y=\sqrt{1-\sin^2y}=\sqrt{1-x^2}$.

For $-1<x<1$, cosine is strictly positive, so division gives $y^{\prime}=1/\sqrt{1-x^2}$.

The endpoints are excluded from this finite derivative formula.

Both copies point to Appendix page 20, Examples D and E, for the arccosine and arctangent proofs; both proofs are now included as Appendix Examples D and E below.