The Tangent Line Shows the Slope

The derivative of a function at a point tells you how steep its curve is exactly there. You find it by drawing the straight line that just touches the curve at that point and measuring that line’s slope.

This picture is part of the full library

Every lesson is open to everyone for 7 days after it is published. This one came out on September 11, 2026: its text stays readable here, and its picture opens with a subscription.

See the plans

The app is free. A subscription opens every lesson, in every school year.

Age 16: open this week

Browse Age 16

In the picture

The blue curve is the graph of a function f(x).
The red dot is the chosen point (a, f(a)) on the curve.
The orange tangent line touches the curve at the red dot and follows its direction there.
The green triangle measures rise over run along the tangent, and that ratio is the derivative f'(a).

Examples

A car’s speedometer shows the derivative of distance travelled with respect to time at that exact instant.
How steep a mountain road is right under your bike’s wheel is the slope of the tangent to the road’s profile at that spot.

Key takeaways

f'(a) equals the slope of the tangent line at x = a.
A positive derivative means the curve is rising, a negative one means it is falling, and zero means a flat tangent, like at the top of a hill.