Chapter 12 — Limits and Derivatives

Chapter 12 of the Class 11 Maths NCERT textbook, "Limits and Derivatives", is an introduction to calculus covering how functions change; derivatives measure the instantaneous rate of change at a point, computed as the limit of average rates of change.

• Limits as the gateway to calculus — The chapter builds calculus on the idea of a limit — the value a function approaches near a point — establishing the foundational concept that every later definition of change relies on.
• Derivatives as instantaneous rates — It defines the derivative as a limit of average rates, capturing how fast a function changes at a single instant, which is the tool behind physical quantities like velocity and acceleration.
• Rules that make derivatives practical — Rather than computing every derivative from first principles, the chapter provides algebra of limits plus product and quotient rules and standard derivatives, so complex functions can be differentiated systematically.

1. 01Limit of f(x) as x→a equals l when both left and right hand limits exist and equal l (denoted lim f(x) = l)
2. 02Derivative f'(a) = lim[f(a+h)−f(a)]/h measures the slope of the tangent to the curve at point a
3. 03Algebra of limits: sums, differences, products, and quotients of limits equal limits of those operations (when denominators ≠ 0)
4. 04Key trigonometric limits: lim(sin x/x) = 1 and lim(1−cos x)/x = 0 as x→0; proven using the sandwich theorem
5. 05Derivative rules: (u+v)' = u'+v', (uv)' = u'v+uv' (product rule), (u/v)' = (u'v−uv')/v² (quotient rule)
6. 06Standard derivatives: (x^n)' = nx^(n−1), (sin x)' = cos x, (cos x)' = −sin x, (tan x)' = sec² x