If you've learned how to find a derivative but still can't answer the question "so what do I actually use this for?", you've only learned half of differentiation. The real power of a derivative isn't the calculation itself — it's that the sign of f'(x) alone tells you exactly what the original function f(x) looks like.
The principle is simple. f'(x) is the instantaneous slope of the graph at that point. If the slope is positive (an uphill), the function is increasing; if it's negative (a downhill), it's decreasing. So what about the point where f'(x)=0 — where the graph momentarily flattens out? If it flattens out after going uphill and then turns back downhill, that point is a peak — a local max. Conversely, if it flattens out after going downhill and then turns back uphill, it's a valley — a local min.
Go one step deeper from here, and you get concavity. If f'(x) tells you "how the slope is changing," then f''(x) (f'(x) differentiated one more time) tells you "how fast the slope itself is changing." If f''(x)>0, the slope is getting bigger and bigger, so the graph is concave up, like a bowl; if f''(x)<0, the slope is getting smaller and smaller, so the graph is concave down, like a mountain. The point where this concavity flips is an inflection point. The sign of f''(x) is also used to confirm whether a point where f'(x)=0 is genuinely an extremum — you can determine this from calculation alone: f''(x)<0 means a local max, and f''(x)>0 means a local min.
All of this really shines in practice when you solve an optimization problem. Questions like "how do I maximize the area of a fenced-in garden with a fixed perimeter?" or "how do I maximize the volume of a box with a limited amount of material?" might seem like they need to be solved by guesswork at first glance, but there's actually a very clear procedure. ① Write the quantity you want (area, volume, etc.) as an expression in a single variable, ② differentiate that expression, and ③ find the point where the derivative becomes 0. For example, differentiate the area A(w)=w(10−w) of a rectangular garden with a 20m perimeter, and you get A'(w)=10−2w — set this to 0 and solve, and you get w=5. Instead of guessing and checking every possible width, one derivative immediately gives you the answer: "the area is maximized when it's a square (5m×5m)."
On our activity page, you can move x with a slider and watch a local max or local min actually appear the moment the sign of f'(x) flips, use the sign of f''(x) to find concavity and inflection points, and finally solve the garden problem above yourself, seeing with your own eyes exactly why "differentiate and set it to 0" is the answer to optimization.