SL 5.1—Limits and derivative concept

A derivative is a local rate of change.

The derivative is the limit of average change as the interval shrinks; it describes the tangent slope at a point.

Example

For f(x)=x², (f(1+h)−f(1))/h=2+h, so f′(1)=2 as h→0.

State the variable and point before differentiating, then interpret the sign and units of the result.

A derivative at one point is not the total change across an interval.

At SL, estimate $\lim_{x\to a}f(x)$ from values approaching $a$ on both sides or from the graph; formal analytic limit methods are not required. The derivative is the limiting gradient and may be written $dy/dx$, $f'(x)$, $dV/dr$ or $ds/dt$, with units of output per unit input. A limit can exist even when the displayed point is missing, because it describes nearby behaviour rather than only $f(a)$.