## Calculus: Early Transcendentals (2nd Edition)

$$f'(a) = \lim_{\Delta x\to 0}\frac{\Delta y}{\Delta x}.$$ The last expression can be interpreted as dividing the small change of the value of the function with the small change of the argument at $x=a$ and the last limit is thus denoted by $$\frac{dy}{dy}.$$
If we denote the change of the value of the function by $\Delta y=f(x)-f(a)$ and the change its argument by $\Delta x=x-a$ we can write $$f'(a) = \lim_{x\to a} \frac{f(x)-f(a)}{x-a} = \lim_{\Delta x\to 0}\frac{\Delta y}{\Delta x}.$$ The last expression can be interpreted as dividing the small change of the value of the function with the small change of the argument at $x=a$ and the last limit is thus denoted by $$\frac{dy}{dy}.$$