Where is the definition of the derivative formula derived from?

Question

I know what the definition of the derivative is , however, I am curious where this comes from mathematically.

Answer 1

So the derivative of a function is its instantaneous slope. And the slope between two points is

$$m=\frac {y_{1}-y_{2}}{x_{1}-x_{2}} $$ The instantaneous slope is conceptually the slope between a point and itself, but we can't quite do this. So instead we must take the slope of the two points: $(x, f (x))$ and $(x+h, f(x+h))$. This yields the slope

$$\frac {f(x+h)-f(x)}{h}$$

Then to simulate the effect of take taking the slope of "a point and itself" we take the limit as $h\to 0$. Yielding the final equation

$$\lim_{h\to 0}\frac {f(x+h)-f(x)}{h}$$