Continuity & Differentiability Chapters Timestamps - 2025 PYQs
[0:00] Introduction and Overview Introduction to differentiability, continuity, and applications of derivatives with focus on short answer questions from 2025 exams. Explanation of the video approach and invitation for viewer queries. [1:00] Proof of Derivative Relation for Given Function Step-by-step proof involving differentiation of a function y = x + (x²+1)^p, calculation of first and second derivatives, and simplification to prove a differential equation. [15:00] Differentiation of Exponential and Implicit Functions Solving problems involving differentiation of functions like y = y^x / x, using logarithmic differentiation and implicit differentiation techniques with detailed steps. [25:00] Increasing Functions and Derivative Conditions Determining values of parameters for which given functions are increasing, by analyzing first derivatives and applying inequalities to find parameter ranges. [35:00] Logarithmic and Trigonometric Differentiation Problems Differentiation of functions involving logarithms and trigonometric expressions, including proving differential equations and using chain and product rules. [45:00] Implicit Differentiation and Related Proofs Solving implicit differentiation problems, including finding dy/dx for complex relations and proving given derivative expressions. [55:00] Continuity and Differentiability at Specific Points Examining differentiability and continuity of piecewise functions at given points by calculating left-hand and right-hand derivatives and limits. [1:05:00] Differentiation of Composite and Inverse Functions Differentiating composite functions such as tan inverse of sin x + cos x, applying chain rule and proving increasing nature of functions over intervals. [1:15:00] Applications of Derivatives in Geometry and Rates Applying derivatives to real-world problems like rate of change of water height in a leaking cylindrical container, using volume formulas and related rates. [1:25:00] Advanced Differentiation Techniques and Proofs Using logarithmic differentiation, product rule, and chain rule to differentiate complex expressions and proving identities involving second derivatives.
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