The derivative as the slope of the tangent

Before any formula, the derivative is a geometric concept: it is the number that measures how steep the curve is at a point. If the graph of a function $f$ rises steeply, the derivative is a large positive number; if it descends gently, it is a small negative number; if it is flat, the derivative is zero.

Definition — The derivative as slope

The derivative of $f$ at the point $x_0$, denoted by $f'(x_0)$ or by $\frac{df}{dx}\big|_{x_0}$, is the slope of the tangent line to the graph of $f$ at the point $(x_0, f(x_0))$.

How does this tangent line arise geometrically? One starts from a secant line and passes to the limit.

From the secant to the tangent: the orange secant passes through $P$ and $Q$ with slope $\Delta f/\Delta x$; as $Q$ approaches $P$ along the curve the secant rotates and tends to the tangent (red), whose slope is the derivative.

The secant line has as its slope the ratio $\Delta f/\Delta x$ between the change in height and the change in the abscissa. As the second point $Q$ slides towards $P$, the secant “rotates” and settles onto the curve: in the limit it becomes the tangent, and its slope is precisely the derivative.

Links

Topics: Derivatives
Concepts: Slope · Derivative · Line · Tangent line
Skills: Interpreting a graph