Trigonometry/Derivative of Inverse Functions

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The inverse functions , etc. have derivatives that are purely algebraic functions.

If then and

.

So

Similarly,

.

If then and

.

So

If then and

.

So

Power series[edit]

The above results provide an easy way to find the power series expansions of these functions.

This is uniformly convergent if so can be integrated term by term. The constant of integration is zero since , so

This is uniformly convergent if so can be integrated term by term. The constant of integration is zero since , so

Note that has no power series expansion about , as it is not defined for and has an infinite derivative when . An expansion about any point in powers of can be found uding Taylor's theorem; it will converge for .