Pick a function type, compare the tangent line and derivative graph side by side, and check whether a slope you compute directly from the definition of a limit lines up exactly with the formula.
A derivative tells you how fast a function is changing at a particular point. On a graph, it's exactly equal to the slope of the tangent line drawn at that point.
Since every function has a different shape, the derivative formula is different for each one too. For example, the derivative of sin x is cos x, and the derivative of eˣ is eˣ itself, unchanged. Formulas like these are all derived by taking the definition of a limit, [f(x+h)−f(x)]÷h, and letting h get arbitrarily close to 0.
Derivatives are widely used any time you need to describe how fast something is changing right now — velocity (the instantaneous rate of change of position), acceleration (the instantaneous rate of change of velocity), and more. Pick a function and compare the graph where the tangent line is drawn against the derivative graph, side by side.