Prerequisites: Definition of Derivative, Derivative Function, Chain Rule

Goal: To visualize the chain rule.

The blue curve represents the graph of f(x)=sin(k x), where k is a constant chosen by the vertical slider. The green line represents the tangent line at the point chosen by the horizontal slider. The red curve represents the derivative function f'(x) of f(x).

  1. What is the amplitude and frequency of f(x)=sin(1.0 x)?
  2. What is the derivative of f(x) = sin(1.0 x)? What is the amplitude and frequency of f'(x)?
  3. What is the amplitude and frequency of g(x)=sin(2.0 x)?
  4. What is the amplitude and frequency of g'(x)? Conjecture the derivative g'(x) of g(x) = sin(2.0 x)?
  5. Without changing anything, predict what you will see in the graphs of the derivatives of sin(3.0 x), sin(4.0 x), and sin(0.2 x). Now adjust the function to check your predictions. How are the function and its derivative related? What can you conclude about the frequency and amplitude of the function and its derivative?
  6. Let f(x) = sin(n x). Using what you know, and what you observed from the graph, what can you say about f'(x)?
  7. Let g(x) = k x so that f(x) = sin(g(x)).
    1. What is g'(x)?
    2. What is f'(x)?
    3. Now let F(x) = f(g(x)). Make a conjecture about the value of F'(x).