The speedometer needle
Your car's trip covers 60 miles in an hour, but the speedometer shows your speed at this exact instant, which changes moment to moment. The derivative is that speedometer: the slope of your distance curve at a single point in time.
Rate of change
The derivative f′(x) measures how fast f changes at x — the slope of the tangent line.
Power rule: for f(x) = xⁿ, f′(x) = n·xⁿ⁻¹. The derivative of a constant is 0.
f(x) = 3x⁴ → f′(x) = 12x³
Simulation · Parabola & tangent slope
y = 1x² · slope at x=1 (the derivative 2a) = 2
The dashed line is the tangent at x=1. Its slope is the derivative 2a — change a and watch the slope change with it.