To compute the derivatives of the inverse trigonometric functions, we will need to simplify composite expressions such as cos(sin −1 x) and tan(sec −1 x). This can be done in two ways:

Simplify cos(sin −1 x) and tan(sin −1 x).

THEOREM 1 Derivatives of Arcsine and Arccosine Derivatives of Arcsine and Arccosine

f (x) = arcsin(x 2 )

THEOREM 2 Derivatives of Inverse Trigonometric Functions

The formulas for the derivatives of the inverse trigonometric functions yield the following integration formulas. Integral Formulas In this list, we omit the integral formulas corresponding to the derivatives of cos −1 x, cot −1 x, and csc −1 x

We can use these formulas to express the inverse trigonometric functions as definite integrals. For example, because sin −1 0 = 0, we have:

Using Substitution

THEOREM 1 Derivatives of Arcsine and Arccosine