The derivative

The derivative measures how fast a function is changing at a single point. Geometrically, it is the slope of the line that just touches the curve there: the tangent.

From secants to the tangent

Pick a point on the curve and a second point a small distance $h$ away. The straight line through the two, the secant, has slope

\frac{f (x + h) - f (x)}{h},

the change in height over the change in width. This is an average rate of change over the little interval. Now let the second point slide toward the first, shrinking $h$ toward zero. The secant pivots until it rests against the curve as the tangent, and its slope becomes the instantaneous rate of change.

A curve with a secant line through two points collapsing onto the tangent line at a single point, illustrating the slope of the tangent as the limit of secant slopes.

The definition

The derivative of $f$ at $x$ is that limiting slope:

f' (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h},

when the limit exists. The quotient inside is $0 / 0$ in the limit, which is precisely why we needed limits first.

A worked example

Take $f (x) = x^{2}$ . Then

\frac{(x + h)^{2} - x^{2}}{h} = \frac{2 x h + h^{2}}{h} = 2 x + h .

As $h \to 0$ this approaches $2 x$ , so $f' (x) = 2 x$ . At $x = 3$ the curve climbs with slope $6$ .

Notation

Newton wrote $f' (x)$ ; Leibniz wrote $\frac{d y}{d x}$ , read "the derivative of $y$ with respect to $x$ ." Both say the same thing: the rate at which the output changes as the input nudges. A function that has a derivative everywhere is called differentiable, and differentiability is a stronger, smoother condition than mere continuity.