Two compositions of inverse trigonometric functions are given:
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
Two compositions of inverse trigonometric functions are given:
(a) Without performing any algebraic differentiation, compare the domains of u and v. State each domain explicitly and explain the reasoning behind your conclusions.
(b) Derive u′(x) and v′(x), simplifying each derivative as much as possible.
(c) Focusing on the interval where both derivatives exist, analyze and discuss how the relative magnitudes of u′(x) and v′(x) change as x increases from 0 to 21. Support your discussion with appropriate inequalities or sign charts.
(d) Summarize, in clear prose, the key structural differences that cause u and v to exhibit the distinct derivative behaviors identified in part (c).