Continuity and Differentiability Class 12 Maths Notes – Formulas & Quick Revision

Continuity and Differentiability Class 12 Calculus ka core hai — chain rule, implicit, logarithmic differentiation sab yahan. Yahan aapko is chapter ke sabse important formulas aur quick revision points free milte hain — NCERT/CBSE syllabus ke hisaab se, boards aur JEE/NEET dono ke liye.

Important Formulas & Concepts — Continuity and Differentiability

• Continuous at a: LHL = RHL = f(a)
• Differentiable ⟹ continuous (converse false)
• Chain rule: dy/dx = dy/du × du/dx
• d/dx(sin⁻¹x) = 1/√(1−x²); d/dx(tan⁻¹x) = 1/(1+x²)
• d/dx(aˣ) = aˣ·ln a; d/dx(logₐx) = 1/(x·ln a)
• Logarithmic differentiation: y = uᵛ → ln y = v·ln u
• Parametric: dy/dx = (dy/dt)/(dx/dt)
• Rolle theorem aur LMV theorem ki conditions

Quick Revision Points

• |x| zero par continuous par differentiable nahi — classic example
• Piecewise functions mein k nikalna common question hai
• x^x type mein log lena zaroori hai
• Second derivative parametric mein dy/dx ko phir t se differentiate karke divide karo
• LMV: f′(c) = [f(b)−f(a)]/(b−a)

FAQ

Derivatives ke formulas kitne hain?

~20 core formulas — PDF ke one-page differentiation sheet mein sab grouped hain.

k nikalne wale questions?

Continuity ki equation banao (LHL = RHL = f(a)) — PDF mein solved pattern hai.

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