the definition of a derivative is to find the slope of a tangent line.....
the alternate form of the derivative is for???
$\displaystyle \lim_{x\to c}=\frac{f(x)-f(c)}{x-c}$ is the alternate form of a derivative.
While $\displaystyle \lim_{\Delta x\to 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}$ is the definition of derivative.
The alternate form will give you the value of the derivative at x=c
So really the definition of a derivative $\displaystyle {f}'(x)$ really maps all of the alternate definitions of a derivative at all x values in the domain of f(x)