math / Calculus / free
Derivatives
See a secant line become a tangent, one small step at a time.
Follow the mechanism1 / 3
Two nearby points define a secant. A finite gap gives an average rate of change between P and Q.
See a secant line become a tangent, one small step at a time.
Two nearby points define a secant. A finite gap gives an average rate of change between P and Q.
The secant is 1.25 away from the tangent. Bring Q closer to P.
Move P. Each tangent slope above becomes one point below, revealing f′(x) = 2x.
Use the pattern f′(x) = 2x before choosing.
This model uses f(x) = x² so the exact tangent slope is known: f′(x) = 2x. The secant is an estimate; the solid tangent is the value it approaches.