The Derivative

A slope needs two points. Bring the second one closer and closer, and the secant becomes the tangent — the slope of the curve at a single point.

Step 1 of 28

— f(x) = x²/2

Speed

f(x) = x²/2, and the point P = (1, 0.5). How steep is the curve exactly at P? A slope needs two points, so start with a second point Q, h to the right.

The secant slope is rise over run between P and Q. Its limit as h → 0 — when it exists — is the derivative f′(x₀), the slope of the tangent at P.

DSAverse

The Derivative

See also

The Derivative

See also