A secant line cuts a curve at two points and measures the average steepness between them. But what if you want the steepness at a single point — how steep is the curve right here?

The line that just grazes the curve at one point, matching its direction there, is the tangent line. Finding its slope is the famous tangent line problem, and solving it is the doorway to calculus.

Let the second point slide in

Here is the beautiful idea. Pin one point P on the curve. Take a second point Q a step h away and draw the difference quotient secant through them. Now slide Q toward P by shrinking h toward 0.

As Q approaches P, the secant line pivots and settles onto the tangent line. Drag h all the way down and watch the secant become the tangent.

This is a limit

We can never set h = 0 outright — the secant would need two points, and at h = 0 both points are the same, giving the undefined fraction \tfrac{0}{0}. Instead we ask what the slope approaches. That is precisely the idea of a limit.

The slope of the tangent line is the limit of the secant slope as the step shrinks to zero:

For f(x) = x^2 the difference quotient simplified to 2x + h. As h \to 0 the h term vanishes, leaving a tangent slope of 2x. At x = 1 that is 2 — and the figure below confirms it.

Watch it explained

Sal Khan shows the tangent slope emerging as the limiting value of secant slopes.