Chapter 5 — Continuity and Differentiability

Chapter 5 of the Class 12 Maths NCERT textbook, "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.

• Continuity versus differentiability — The chapter makes precise what it means for a function to have no breaks (continuity) and to have a well-defined tangent (differentiability), then shows the two are linked but not the same — every differentiable function is continuous, yet |x| is continuous without being differentiable at 0.
• Differentiating harder functions — Building on Class 11 rules, the chapter equips students to differentiate composite functions through the chain rule, and extends the toolkit to inverse trigonometric, exponential and logarithmic functions so that a much wider range of expressions can be handled.
• Special differentiation techniques — For awkward forms it introduces dedicated methods — logarithmic differentiation for power-tower expressions like [u(x)]^v(x), parametric differentiation when x and y depend on a parameter, and second-order derivatives that measure how the rate of change itself changes.

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.