Three Forms, One Sign Rule, and a Radical You Can Measure

You'll learnto get all six inverse trigonometric derivatives from three memorised forms plus one rule about negatives, to see the triangle the radical is a side of, and to catch a swapped form by the shape of its graph instead of by re-deriving it.

Six derivatives is a lot to hold. Three is not — and the other three are the same three with a minus sign, because a co-function is its partner reflected, and reflecting a curve flips every slope it has. That is the whole memory problem solved. What is left is knowing which family a function belongs to, and that is a question you can answer by looking: arcsine lives only between −1 and 1 and stands up vertically at both ends, so its derivative has to run away there; arctangent accepts every number and flattens out, so its derivative has to stay small and never break. A student who remembers those two pictures cannot swap the two forms, which is the error this topic is really about.

8 STEPS · 6 QUICK CHECKS · THREE FORMS + ONE SIGN RULE · THE RADICAL IS A SIDE OF A TRIANGLE · v1

three families ⇄ radical · rational · absolute value

Before you start

What you're looking at

Six derivatives that look like six things to memorise, laid out as what they actually are: three forms and their three mirrors. Every curve here is plotted from the function itself, and every slope is measured off that plotted curve rather than typed in.

The question

Why does arcsine get a radical and arctangent get a sum — and how do you tell, in two seconds, that you have not swapped them?

Watch for

The triangle in step 3, where the radical turns out to be a side you can measure, and step 7, where claiming a positive derivative for arccosine draws a rising tangent across a curve that falls.

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