How to remember the derivatives of inverse trigonometry function? I remember the derivatives of trig functions  by naming 3x basic right triangles in a specific way and using ONE simple multiplication.  Just wondering if there are  similar approaches  to remember the derivatives of inverse trig function with the assisting of some basic triangles?

Procedures $\to$

*

*Draw a right triangle (1, $\cos x,sinx$).Label its sides accordingly as shown in Fig.1.


*Put the hypotenuse label ("1" in this case)  in side the triangle , name this triangle as "1" triangle.


*Here is the only rule:  |X'| = |Central value  *   Y|, where X and Y are the  two functions on the NONE "1" sides.  In this case, the central value is "1", and the two functions on the NONE "1" sides are $sinx$ and $cos x$.
$           $|$sinx'| =|1* cos x$| ,  $|cosx'|= |1*sinx|$ (Fig.1)
Similarly, divide each side of "1"triangle by $sinx$ and $cosx$ to obtain "$sec"  and  “$cscx" triangles in Fig. 2 and Fig.3.
$|secx'| =|secx*tanx|$,  $|tanx'|= sec*secx$  (Fig.2)
$|cscx'| =|cscx*cotx|$,  $|cotx'|= cscx*cscx$  (Fig.3)
(Need to apply negative signs to get values for $cosx', cotx' $and $cscx'$).
 A: Here's a slight re-packaging of your mnemonic.
We start with the figure I call the Fundamental Trigonograph (the inspiration for my avatar!), whose segment-lengths correspond to the trig values associated with (acute) angle $\theta$.

Note that the "$1$" segment separates "ordinary" segments ($\sin$, $\tan$, $\sec$) from "complementary" segments ($\cos$, $\cot$, $\csc$).
Then, with no motivation whatsoever, we augment the figure with two right triangles having legs $\sec$ and $\csc$, and hypotenuses parallel to the "$1$" segment. (I've suppressed the "$1$" label to reduce visual clutter, but extended it to highlight the separation between "ordinaries" and "complementaries".)
One readily determines the new side lengths to be $\sec\tan$, $\sec^2$, $\csc\cot$, $\csc^2$; by interesting coincidence, lengths $\sin$ and $\cos$ appear in the lower-left quadrant in an obvious way. I've attached negative signs to the new segments on the "complementary" side of the augmented figure ... um ... just because. :)

With this, the rule is something like:
$$\text{A function and its derivative are "natural" perpendicular pairs on the same side of "$1$".}$$
That is,

*

*$\tan$ "naturally" pairs with $\sec^2$, and $\sec$ with $\sec\tan$. The pairing of the original $\sin$ segment with the augmenting $\cos$ segment is slightly contrived, but it works.

*Likewise, $\cot$, $\csc$, and $\cos$ "naturally" pair with $-\csc^2$, $-\csc\cot$, and $-\sin$.


Is this the best presentation of the mnemonic? I'm not sure ---it's just what occurred to me in the moment--- but I kinda like it. I especially like that it builds on the Fundamental Trigonograph, which absolutely belongs in every student's tool belt. (See, for instance, my still-drafty note "(Almost) Everything You Need to Remember about Trig, in One Simple Diagram" (PDF link via tricochet.com). I may need to make an addition!)
