How to remember the derivatives of inverse trigonometry function?

Question

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 $\to$

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

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

3. 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 $cscx'$).

Answer 1

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 $-\sin$.

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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!)