Mathematics

Problem

A function is defined by

p(x)=\arcsin\bigl(\arcsin(x)\bigr),

where

p

is considered only on its maximal domain. (a) Determine the exact domain of

p

and explain how you obtain it. (b) Calculate

p'(x)

for all

x

in the domain found in part (a). Express your result in simplest form. (c) Evaluate

p'\!\bigl(\tfrac12\bigr)

exactly. (d) Interpret the sign of

p'(x)

on the interior of its domain. What does this indicate about the monotonic behavior of

p

? Provide reasoning in complete sentences.