A derivative tells you the rate of change (the gradient) of every point on the curve/line.
For instance take the linear function (a straight line) y = 2x. Rise / Run will show you the gradient equals 2 on any point of the line. The derivative of y (y'=2) also tells you that the gradient will always equal 2 at any point of x.
On a non-linear function however y=x^2 +2x, the gradient changes on every point of the curve and the rise/run method will not work, but the derivative will and it will give you an equation that can tell you the gradient of the function at any point x, in this case the derivative y' = 2x + 2. At point x=0, y'=2 therefore the gradient at x=0 is 2. At point x=7.5, y'=17 therefore the gradient at x=7.5 is 17 etcetera.
