AI Tutor
Back to The chain rule

Exercise 11

Avg difficulty 2.5

Consider the basic functions f(x)=x3f(x) = x^3 and g(x)=sin⁡(x)g(x) = \sin(x).

  1. Let h(x)=f(g(x))h(x) = f(g(x)). Find the exact instantaneous rate of change of hh at the point where x=π4x = \frac{\pi}{4}.

  2. Which function is changing most rapidly at x=0.25x = 0.25: h(x)=f(g(x))h(x) = f(g(x)) or r(x)=g(f(x))r(x) = g(f(x))? Why?

  3. Let h(x)=f(g(x))h(x) = f(g(x)) and r(x)=g(f(x))r(x) = g(f(x)). Which of these functions has a derivative that is periodic? Why?

Boelkins, Active Calculus, §2.5 The chain rule, Exercise 11

AI Tutor

Adapted from Active Calculus © 2012–2025 Matthew Boelkins. Licensed under CC BY-SA 4.0. Source: Active Calculus. Imported at revision dc86ba3.