Consider the function f(x)= ( (2x) ) restricted to the open interval .
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
Consider the function
f(x)=arcsin(arctan(2x))
restricted to the open interval (−21,21).
(a) Determine the largest open interval contained in (−21,21) on which f is differentiable. State all domain considerations that justify your answer.
(b) Derive an explicit formula for f′(x) on the interval found in part (a). Express your answer as a single simplified fraction in terms of x.
(c) Evaluate f′(0) exactly.
(d) Using the derivative obtained in part (b), determine the set of x–values in the interval from part (a) for which f is increasing. Justify your conclusion algebraically.