The derivative represents the instantaneous rate at which one variable changes with respect to another. Geometrically, it gives the gradient of the tangent to a curve; the gradient function gives this gradient at every point.
For a function , the average rate of change from to is
This is the gradient of a secant line through two points on the curve. As approaches zero, the second point approaches the first, and the secant approaches the tangent. The resulting instantaneous rate of change is
In IB Mathematics: Applications and Interpretation SL 5.1, this limit is understood informally using graphs or tables; formal analytic limit methods are not required.
This distinction matters: an average rate of change describes change across an interval, while a derivative describes change at one instant. If , the tangent is horizontal, although this alone does not prove that has a maximum or minimum.
| Representation | Meaning |
|---|---|
| (f'(a)) | The gradient or instantaneous rate of change at |
| (f'(x)) | A function giving the gradient of for every value of |
| or | The original function is increasing or decreasing, respectively |
| The tangent is horizontal at |
For example, suppose the height of an object is
where height is measured in metres and time in seconds. Its derivative is
At ,
Therefore, the object’s height is increasing at . The units of a derivative are always “output units per input unit.” A positive derivative shows that height is increasing, while a negative derivative shows that it is decreasing. The magnitude gives the steepness of that change.
When asked to interpret a derivative, give the value, units, and contextual meaning. A common misconception is that (f'(a)) gives the function’s value; it gives the function’s gradient or rate of change at that point. In Math AI, a GDC may be used on every paper, but show the derivative or rate-of-change setup before giving the calculator value.
