Difference quotient and derivative

How does one compute the slope of the tangent? Starting from the slope of the secant and passing to the limit: the secant is determined by two points, but by letting the second slide towards the first the ratio of the changes tends to a well-defined value.

Definition — Difference quotient and derivative

The difference quotient of $f$ at $x_0$ with increment $\Delta x$ is: $\frac{\Delta f}{\Delta x} = \frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}.$ The derivative is the limit of this quotient as $\Delta x\to 0$: $f'(x_0) = \lim_{\Delta x\to 0}\frac{\Delta f}{\Delta x} = \lim_{\Delta x\to 0}\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}.$

$\Delta x$ is the horizontal displacement between the two points, $\Delta f$ the vertical one: the ratio $\Delta f/\Delta x$ is the slope of the secant. In the limit $\Delta x\to 0$ the secant becomes the tangent and the ratio becomes the derivative.

Explore the construction in the simulation: drag the slider $h$ towards $0$ and watch the secant turn into the tangent.

Drag $x_0$ and push $h$ towards $0$: the secant (red) approaches the tangent (dashed) and the difference quotient $\Delta y/\Delta x$ tends to $f'(x_0)$.

Links

Topics: Derivatives
Concepts: Derivative · Limit · Difference quotient
Skills: Calculating limits · Interpreting a graph