Chapter 5 — Continuity and Differentiability

Class 12 Maths Chapter 5 — Continuity and Differentiability — covers the formal definitions of continuity and differentiability, the chain rule, derivatives of inverse trigonometric, exponential, and logarithmic functions, logarithmic differentiation, parametric differentiation, and second-order derivatives.

Chapter 5 of NCERT Class 12 Mathematics Part I extends Class 11 differentiation to introduce continuity and differentiability rigorously. A function is continuous at a point c if its limit equals its value there; every differentiable function is continuous, but not conversely (f(x) = |x| is continuous yet not differentiable at 0). The chapter introduces the chain rule for composite functions, implicit differentiation, derivatives of inverse trigonometric functions (sin⁻¹x, cos⁻¹x, tan⁻¹x), exponential and logarithmic functions (d/dx(eˣ) = eˣ; d/dx(log x) = 1/x), logarithmic differentiation for functions of the form [u(x)]^v(x), parametric differentiation, and second-order derivatives.

1. 01A function f is continuous at c if lim(x→c) f(x) = f(c); the greatest integer function [x] is discontinuous at every integer.
2. 02Every differentiable function is continuous, but the converse is false — f(x) = |x| is continuous at 0 but not differentiable there.
3. 03The chain rule states df/dx = (dv/dt)·(dt/dx) for composite f = v∘u with t = u(x), enabling differentiation of functions like sin(x²).
4. 04Derivatives of inverse trig functions: d/dx(sin⁻¹x) = 1/√(1−x²), d/dx(cos⁻¹x) = −1/√(1−x²), d/dx(tan⁻¹x) = 1/(1+x²).
5. 05The natural exponential function satisfies d/dx(eˣ) = eˣ, and d/dx(log x) = 1/x; logarithmic differentiation handles [u(x)]^v(x) forms.
6. 06For parametric equations x = f(t), y = g(t), dy/dx = g′(t)/f′(t) provided f′(t) ≠ 0; the second-order derivative d²y/dx² is the derivative of dy/dx.