Establish the following formula for the derivative of is correct,

For let

Then we know

Therefore, by Theorem 6.7 (p. 252 of Apostol) we have,

Where we use the trig identity in the final line. Then, since we have,

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Stumbling Robot

A Fraction of a Dot
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Establish the formula for the derivative of *arccsc x*

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Establish the following formula for the derivative of is correct,

For let

Then we know

Therefore, by Theorem 6.7 (p. 252 of Apostol) we have,

Where we use the trig identity in the final line. Then, since we have,

In the final line, why did the -1 come back?

We can take the positive or negative square root. However, the restricted csc function, for which arccsc is the inverse, is decreasing everywhere it is defined. The inverse of a decreasing function is decreasing. So arccsc is decreasing everywhere, and its derivative must be negative everywhere.