$$\begin{aligned} f'(x) &= \lim_{h\to0}\frac{f(x+h) - f(x)}{h}\\ &= \frac{df}{dx} \end{aligned}$$
In words:
So the derivative is a small change in the outcome of a function divided by a small change in the argument of that function.
Let us define the following function:
$$y = 30*sin(x\frac{\pi}{50})$$
Then we can write the above formula for the derivative as:
You can use the following slider to change the point at which you want to take the derivative
You can use the following slider to influence $h$. Notice how for every point, when the value of $h$ goes to zero, then the calculated derivatives approaching from the left and the right converge to the same value. Our function is derivable.
(If you move this slider too fast the visuals can't follow, so be easy with it)
Let us define the following function:
$$y = abs(30*sin(x\frac{\pi}{50}))$$
Then we can write the above formula for the derivative as:
You can use the following slider to change the point at which you want to take the derivative
You can use the following slider to influence $h$. Notice how for $x=0$, when the value of $h$ goes to zero, then the calculated derivatives approaching from the left and the right do NOT converge to the same value. Our function is NOT derivable.
(If you move this slider too fast the visuals can't follow, so be easy with it)