Average rate of change and the derivative at a point

The average rate of change is the slope of the line joining the two ends of an interval¹. We compute it for $f(x) = x^2$ on the interval $[1, 2]$.

That $3$ is precisely the slope of the line through the two points $(1, 1)$ and $(2, 4)$, the secant. That line is $y = 3x - 2$.

Shrinking the interval

What happens to the slope of the secant as the right end of the interval approaches $1$? We compute the average rate of change over $[1, 1 + h]$.

The derivative at a point

The value of that limit is the derivative at $x = 1$, written $f'(1)$.

The derivative at a point is the slope of the tangent there. The line through $(1, 1)$ of slope $2$ is $y = 2x - 1$, and on the graph it touches the parabola at a single point. A secant crosses at two points; a tangent merely touches.

A quantity of an interval and a quantity of a point

On to the derivative function

The same computation at a general $x = a$ gives $\dfrac{(a + h)^2 - a^2}{h} = 2a + h$, which approaches $2a$ as $h \to 0$. Collapsing the interval is exactly the operation that produces the derivative.

Since the computation works at every $x$, the derivative at a point becomes a function of $x$. For $f(x) = x^2$ it is $f'(x) = 2x$, called the derivative function. Putting $x = 1$ gives $2$ and $x = 2$ gives $4$, matching the way the parabola steepens toward the right.

The parabola on the graph is $y = x^2$, the line through the two points is the secant, the line touching the parabola is the tangent, and the large dots are $(1, 1)$ and $(2, 4)$.

1. Derivative, Wikipedia