Conceptual Dependency Math Example 1

Solution

1. 1
   Level 0 (most basic): real numbers, functions, limits.
2. 2
   Level 1 (depends on Level 0): the derivative as $\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$.
3. 3
   Level 2 (depends on Level 1): the limit definition of $\sin$ and $\cos$; the fundamental limits $\lim_{h\to 0}\frac{\sin h}{h}=1$ and $\lim_{h\to 0}\frac{\cos h - 1}{h}=0$.
4. 4
   Level 3: combining the above to compute $(\sin x)' = \cos x$.