Class 12 Math Continuity and Differentiability lect-03

📌 Class 12 Maths | Chapter 5: Continuity & Differentiability (Lecture 03) — Advanced Questions & Differentiability Rules 📐⚡Welcome back to Run Academy! 🎓 In Lecture 03 of Continuity and Differentiability, we take your calculus preparation to the next level! Building on the fundamentals from Lect 01 & 02, this session focuses on checking differentiability at a point, Left Hand Derivative (LHD) vs. Right Hand Derivative (RHD), finding unknown parameters ($a, b, k$), and solving high-yield Board Exam & JEE questions! 🚀📚 WHAT YOU WILL LEARN IN THIS LECTURE:🔹 1. Concept & Condition for Differentiability• 📈 Geometric Meaning: Smooth curves vs. sharp corners (kinks) and vertical tangents.• ⚖️ Mathematical Condition: A function $f(x)$ is differentiable at $x = c$ if:$$\text{LHD} = \text{RHD} \implies \lim_{h \to 0} \frac{f(c-h) - f(c)}{-h} = \lim_{h \to 0} \frac{f(c+h) - f(c)}{h}$$🔹 2. Continuity vs. Differentiability Relationship• 💡 Key Theorem: "Every differentiable function is continuous, but every continuous function is NOT necessarily differentiable!"• 🔍 Classic Example: Breakdown of $f(x) = \vert{}x\vert{}$ at $x = 0$ (Continuous but non-differentiable due to a sharp corner).🔹 3. Differentiability of Piecewise & Modulus Functions• ✂️ Modulus & Greatest Integer Functions: How to test differentiability for $\vert{}x - a\vert{}$ and $[x]$.• 🧩 Finding Unknown Constants ($a, b, k$): Step-by-step method using combined conditions of both Continuity AND Differentiability.🔹 4. Standard Differentiation Rules & Tricks• 📐 Algebra of Derivatives: Product Rule, Quotient Rule, and Chain Rule applications.• 🎯 Board & JEE Level Problem Solving: High-probability PYQs solved step-by-step!