Sliders are a pretty good representation to show function composition. Can they help us understand the chain rule as well?[br][br]The horizontal arrows show the rate of change of f, g and g∘f.
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).[br][br]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.[br][br]
If we nudge [math]x[/math] very slightly, what could we call that change?
What could we call the very small change in [math]f[/math] caused by the small change in [math]x[/math]?
What could we call the very small change in [math]g[/math] caused by the small change in [math]f[/math]?
If [math]g=sin\left(f\right)[/math], rewrite the small change [math]dg[/math] in terms of [math]f[/math].
[math]d\left(sin\left(f\right)\right)[/math]
Evaluate [math]d\left(sin\left(f\right)\right)[/math].
[math]d\left(sin\left(f\right)\right)=cos\left(f\right)df[/math]
If [math]f=x^2[/math], rewrite the equation above, but write [math]f[/math] in terms of [math]x[/math]
[math]d\left(sin\left(x^2\right)\right)=cos\left(x^2\right)d\left(x^2\right)[/math]
Rewrite the equation again, evaluating [math]d\left(x^2\right)[/math]
[math]d\left(sin\left(x^2\right)\right)=cos\left(x^2\right)2xdx[/math]
Now divide both sides by [math]dx[/math] and write your final equation
[math]\frac{d}{dx}\left(sin\left(x^2\right)\right)=cos\left(x^2\right)2x[/math]