The difference quotient (f(x+h) − f(x))/h is the average rate of change over a small interval. The derivative is its limit as h → 0.
Drag the point along the curve
y = x·xat x = 1.5: slope ≈ 3
Difference quotient
The slope of the secant line through two points on the curve, h apart. Let h → 0 and the secant becomes the tangent — that's the derivative.
Your turn
For f(x) = x², form the difference quotient, simplify, then let h → 0.
Watch out
Don't set h = 0 before simplifying — you get 0/0. Cancel the h in the denominator first, then take the limit.
Derivative