DIFFERENTIABILITY - Mathematical Definition

Duration: 6 min

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Module outline

  1. General Intelligence & Reasoning: Analogy, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Series (Number and Letter Series) (Numerical Relations and Reasoning), Coding Decoding, Blood Relations, Directions (Direction Test), Ranking, Order and Sequence, Logical Sequence Of Words, Syllogisms, Venn Diagram, Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Selection Decision Table (Decision Making), Data Sufficiency, Puzzle Solving (Puzzles), Data Arrangement, Cubes & Dices, Calendar, Clock, Arithmetic, Number System, HCF LCM, Verbal Analogy (Word Based Analogy), Visual Recognition and Figure Completion, Paper Folding and Cutting, Transformation of Shapes, Clocks and Calendars, Direction Sense, Cubes and Dice, Patterns in Two-Dimensional Figures, Seating Arrangements, Analogy and Classification, Logical Deduction and Induction, Alphabet and Word Tests, Mathematical Operations, Series Completion, Simplification and Approximation, Coding and Decoding, Numerical Reasoning, Assembling and Grouping of Shapes, Three-Dimensional Patterns and Visualization, Vocabulary, Number System, Averages, Mixtures and Alligation, Simple and Compound Interest, Percentages, Algebra, Speed, Time and Distance, Seating and Arrangement Puzzles, Traverse Surveying, Order, Ranking and Sequence, Constraint Puzzles, Syllogisms and Venn Diagrams, Reading Comprehension, Critical Reasoning, Permutations and Combinations, Narrative Sequencing, Project Monitoring, Resources and Safety, Profit, Loss and Discount, Ratio and Proportion, Time and Work, Mensuration, Data Interpretation, Powers, Surds and Logarithms, Sequences and Series, Articles and Determiners, Sentence Completion, Data Sufficiency, Blood Relations, Simple Interest and Compound Interest, Mixture and Alligation, Sequence and Series (Series), Speed Time and Distance, Algebra, Profit and Loss, Powers and Exponents (Surds and Indices), Permutation and Combination
  2. General Awareness: Physics, Chemistry, Biology, History, Geography, Polity, Indian Economy, Environment, Static General Knowledge, GK- Current Affairs, Artificial Intelligence, MS Office Applications, Terms & Abbreviations, Keyboard Shortcuts, Evolution of Indian Banking (Banking & Financial Awareness), Emerging Tech Trends
  3. Engineering Mathematics: Set Theory, Relations, Functions, Graph Theory, Group Theory, Propositional and Predicate Logic, Linear Algebra, Calculus, Probability, Statistics, Signal Space and Optimum Detection, Probability, Statistics
  4. Digital Logic: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation, Floating Point Rep, Number Systems and Codes, Registers and Counters, Boolean Algebra and Logic Gates, Microprocessors and Microcontrollers, Logic Families and CMOS Logic, Microprocessors, Logic Gates and Boolean Algebra, Minimization of Boolean Functions, Timing, Hazards and Testing, Logic Families, Combinational Logic Design, Arithmetic Circuits, Waveform Generators and Timer Circuits, Semiconductor Memories and Programmable Logic Devices, Combinational Logic Circuits, Latches and Flip-Flops, Finite State Machines, Sequential Circuits and State Machines, Minimization of Boolean Functions, Arithmetic Circuits, Latches and Flip-Flops, Number Systems and Codes, Registers and Counters
  5. Computer Organisation & Architecture: Basics of COA, Cache Memory Organization, Input Output Organisation, Pipelining, Instr Formats & Modes, Control Unit Design, Programming the Basic Computer, Register Transfer and Microoperations, Semiconductor Memories and Programmable Logic, Processor Organization and Instructions, Memory Hierarchy and Input/Output, Registers and Counters
  6. Programming & Data Structures: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling, Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing, HTML Basics, OOPS with C++, Java, Object Oriented Technology
  7. Databases: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, File Organization & Indexing, Relational Algebra, SQL, Relational Calculus, Transaction Management, Concurrency Control, Database Recovery, Database Security & Authorization
  8. Algorithms: Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos
  9. Theory of Computation: Introduction to TOC, Deterministic FA (DFA), Non-Deterministic FA, Regular Expressions, Grammar, Regular Language Properties, Moore & Mealy Machines, Pushdown Automata & CFG, Turing Machines, Complexity Theory
  10. Compiler Design: Intro to Compilers, Lexical Analysis, Grammar & CFG, Syntax Analysis: Top-Down, Syntax Analysis: Bottom-Up, Semantic Analysis & SDT, Intermediate Code Gen, Code Optimization, Run Time Environment
  11. Operating System: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, Disc Scheduling, File Management, Security, Windows OS, Distributed Systems, Unix OS, Linux OS
  12. Computer Networks: Introduction to CN, DLL: Access Control, DLL: Flow Control, DLL: Error Control, DLL: Framing, Data Link Layer - Ethernet, Net Layer: IPv4 & Proto, Net Layer: IP Addressing, Net Layer:Routing Protocol, Transport Layer Services, TL: Congestion & UDP, Application Layer, Hardware Basics, Data Communication, Website, Browser & Services, Web Design & Development, Network Security, Cyber Threats and Malware Attacks, Cyber Safety and Security
  13. Full Length Mock Tests:
  14. Sectional Tests:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This educational video provides a focused lecture on the concept of differentiability in calculus, presented by instructor Yash Jain Sir from Knowledge Gate. The session begins with a formal mathematical definition of differentiability at a specific point, establishing the necessary limit condition. A central theme of the lecture is the logical relationship between differentiability and continuity. The instructor explicitly states that while differentiability guarantees continuity, the reverse is not true. To substantiate this claim, he performs a detailed analysis of the absolute value function, f(x) = |x|, demonstrating through limit calculations that a function can be continuous at a point yet fail to be differentiable due to a sharp corner or cusp. The lecture effectively bridges the gap between abstract definitions and practical application.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video opens with a title card featuring the word CALCULUS surrounded by various mathematical formulas. The instructor, identified as Yash Jain Sir from Knowledge Gate, appears in front of a whiteboard titled Differentiability -> Mathematical Definition. He writes the formal definition: a function f: R -> R is differentiable at x=a if f'(a) exists. He expands this to show the limit definition: lim h->0 [f(a+h) - f(a)] / h must exist. Below this, he lists two critical implications: (1) if a function is differentiable, it is continuous, and (2) if a function is continuous, it is not necessarily differentiable, indicated by a crossed arrow. He underlines x=a and f'(a) to emphasize the point of interest.

  2. 2:00 – 5:00 02:00-05:00

    The instructor moves to a concrete example to illustrate the second point. He writes f(x) = |x| and defines it piecewise as x for x >= 0 and -x for x < 0. He asserts that the function is continuous at x=0 but not differentiable. He sets up the derivative limit at x=0: lim h->0 [f(0+h) - f(0)] / h. He splits this into Left Hand Limit (LHL) and Right Hand Limit (RHL). He calculates the RHL as lim h->0+ h/h = 1 and the LHL as lim h->0- -h/h = -1. He draws a V-shaped graph of |x| on the right side of the board to visualize the sharp corner at the origin.

  3. 5:00 – 5:44 05:00-05:44

    The instructor concludes the proof by highlighting that since the LHL (-1) is not equal to the RHL (1), the limit does not exist, and thus the function is not differentiable at x=0. He writes LHL != RHL in red ink to emphasize the inequality. He also writes LD = RD with a checkmark to show the general condition for differentiability, contrasting it with the current case where LD != RD. He circles the limit expression and the result -1 to draw attention to the specific values. The video concludes with a black screen displaying THANKS FOR WATCHING.

The lecture successfully demystifies the relationship between continuity and differentiability. By starting with the rigorous limit definition and immediately applying it to a classic counterexample, the instructor provides a clear, visual, and mathematical proof that continuity is a prerequisite but not a guarantee for differentiability. The step-by-step calculation of one-sided limits for the absolute value function serves as a practical template for students to analyze other non-differentiable points, reinforcing the concept through calculation of one-sided limits.

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