The derivative of f at x is the limit of the difference quotient — it measures instantaneous rate of change.
f'(x) = lim[h→0] [f(x+h) − f(x)] / h
Also written: dy/dx, Df(x), ẋ (time derivatives)
Geometrically, f'(a) is the slope of the tangent line to the curve y = f(x) at x = a.
Worked Example — Derivative of f(x) = x²
Write the limit: lim[h→0] [(x+h)² − x²] / h
Expand: [(x² + 2xh + h²) − x²] / h
Simplify: [2xh + h²] / h = 2x + h
Take the limit as h→0: f'(x) = 2x ✓
Differentiability
f is differentiable at a if f'(a) exists. Differentiability implies continuity (but NOT vice versa). f is NOT differentiable at corners, cusps, or vertical tangents.