Chapter 5 — Continuity and Differentiability — is one of the most important chapters carrying 8-10 marks. Master derivatives of composite, implicit, parametric, and logarithmic functions, plus Rolle’s and Mean Value theorems.
Key Concepts
Continuity
f(x) is continuous at x = a if: lim(x→a) f(x) = f(a)
i.e., LHL = RHL = f(a)
i.e., LHL = RHL = f(a)
Differentiation Rules
Chain Rule: d/dx[f(g(x))] = f'(g(x)) · g'(x)
Product Rule: d/dx[uv] = u’v + uv’
Quotient Rule: d/dx[u/v] = (u’v − uv’)/v²
Standard Derivatives:
d/dx(xⁿ) = nxⁿ⁻¹ | d/dx(eˣ) = eˣ | d/dx(aˣ) = aˣ ln a
d/dx(ln x) = 1/x | d/dx(sin x) = cos x | d/dx(cos x) = −sin x
d/dx(tan x) = sec²x | d/dx(sec x) = sec x tan x
d/dx(sin⁻¹x) = 1/√(1−x²) | d/dx(tan⁻¹x) = 1/(1+x²)
d/dx(cos⁻¹x) = −1/√(1−x²)
Product Rule: d/dx[uv] = u’v + uv’
Quotient Rule: d/dx[u/v] = (u’v − uv’)/v²
Standard Derivatives:
d/dx(xⁿ) = nxⁿ⁻¹ | d/dx(eˣ) = eˣ | d/dx(aˣ) = aˣ ln a
d/dx(ln x) = 1/x | d/dx(sin x) = cos x | d/dx(cos x) = −sin x
d/dx(tan x) = sec²x | d/dx(sec x) = sec x tan x
d/dx(sin⁻¹x) = 1/√(1−x²) | d/dx(tan⁻¹x) = 1/(1+x²)
d/dx(cos⁻¹x) = −1/√(1−x²)
Special Differentiation Techniques
Implicit: Differentiate both sides w.r.t. x, collect dy/dx terms
Example: x² + y² = 25 → 2x + 2y(dy/dx) = 0 → dy/dx = −x/y
Parametric: If x = f(t), y = g(t), then dy/dx = (dy/dt)/(dx/dt)
Logarithmic: Take ln of both sides, then differentiate
Used for: xˣ, x^(sin x), (sin x)^(cos x) type functions
Example: x² + y² = 25 → 2x + 2y(dy/dx) = 0 → dy/dx = −x/y
Parametric: If x = f(t), y = g(t), then dy/dx = (dy/dt)/(dx/dt)
Logarithmic: Take ln of both sides, then differentiate
Used for: xˣ, x^(sin x), (sin x)^(cos x) type functions
Second Order Derivatives
d²y/dx² = d/dx(dy/dx) — differentiate the first derivative again
Rolle’s Theorem
If f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then there exists c ∈ (a,b) such that f'(c) = 0
Mean Value Theorem (Lagrange’s)
If f is continuous on [a,b] and differentiable on (a,b), then there exists c ∈ (a,b) such that:
f'(c) = [f(b) − f(a)]/(b − a)
f'(c) = [f(b) − f(a)]/(b − a)
Quick Revision Points
- Continuity: LHL = RHL = f(a)
- Every differentiable function is continuous, but not vice versa (|x| at x=0)
- Chain rule for composite; log differentiation for variable exponents
- Parametric: dy/dx = (dy/dt)/(dx/dt)
- Rolle’s: f(a)=f(b) → f'(c)=0; MVT: f'(c) = [f(b)−f(a)]/(b−a)
Chapter Navigation
Previous: Determinants Class 12 Notes
Next: Application of Derivatives Class 12 Notes
Related Chapters in Class 12 Maths
- Application of Derivatives Class 12 Notes
- Integrals Class 12 Notes
- Differential Equations Class 12 Notes
Practice What You Learned
Test yourself with our JEE Main Mock Test Set 1 to see how well you’ve mastered the concepts.