Visualization of Chain Rule
Sliders are a pretty good representation to show function composition. Can they help us understand the chain rule as well?
The horizontal arrows show the rate of change of f, g and g∘f.
Inspiration: the excellent 3 blue 1 brown Essence of Calculus Video (Start around 9:00)
With a partner, answer the following questions. Please check each answer before moving on to the next question. You may want to reference the video if you are feeling stuck (start around 9:00).
If you changed the functions f(x) and g(x), please use the circular arrow in the top right corner to reset back to original functions before answering the following problems.
If we nudge very slightly, what could we call that change?
What could we call the very small change in caused by the small change in ?
What could we call the very small change in caused by the small change in ?
If , rewrite the small change in terms of .
Evaluate .
If , rewrite the equation above, but write in terms of
Rewrite the equation again, evaluating
Now divide both sides by and write your final equation