What Is a Derivative?

The problem: slope needs two points

The slope of a straight line is rise over run, and you need two points to measure it. A curve’s steepness changes from point to point, so what could “the slope at P” even mean when there is only one point?

The trick is to cheat with a second point Q a small distance $h$ away, measure the slope of the line through both (a secant line), and then slide Q towards P. Watch what happens to that slope:

As $h$ shrinks, the secant slope settles on a single number. That number is the derivative, and the line it settles into is the tangent line, the one that touches the curve at P and points along it. In symbols:

On ahaboo, you can work through visual explainers on topics like how tax brackets set your rate and how photosynthesis unfolds inside a leaf.

Why it matters: rates of change

If $s(t)$ is your position at time $t$, the secant slope $\frac{s(t+h)-s(t)}{h}$ is your average speed over the next $h$ seconds, and the derivative $s'(t)$ is the speed on the speedometer at that instant. The same idea gives marginal cost in economics, reaction rates in chemistry and growth rates in biology.

From definition to rules

Nobody computes limits every time. Apply the definition once to $x^2$:

Do this for general powers, products and compositions and you get the shortcut rules: the power rule, product rule, quotient rule and chain rule, all collected on derivative rules. The derivative calculator applies them for any function and names each one.

When there is no derivative

The limit has to exist and be the same from both sides. At a sharp corner, like $|x|$ at 0, the secant slope is −1 from the left and +1 from the right, so there is no single tangent and no derivative. The same happens at jumps and at points where the tangent is vertical.