21 Differentiation 1 From First Principles – Advanced Higher Maths Lessons MrThomasMaths SQA

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Here is Advanced Higher Maths, Chapter 2 – Differentiation • Lesson 1 of 17: Differentiation From First Principles • A special thanks to Grace, Ryan and Connor for kindly providing me with a unique musical intro… • 130 videos have now been uploaded to guide you through the Advanced Higher Maths course and each video includes an explanation along with plenty of worked examples to help you gain a solid understanding. Here is a list of both the chapters and the individual lessons to allow you to target specific outcomes. • Enjoy! • CHAPTERS • 1. Partial Fractions • 2. Differentiation • 3. Integration • 4. Differential Equations • 5. Binomial Theorem • 6. Sequences Series • 7. Proof by Induction • 8. Functions Graphs • 9. Applications of Calculus • 10. Gaussian Elimination Matrices • 11. Vectors • 12. Complex Numbers • 13. Euclidean Algorithm • 14. Proofs • LESSONS • 1. PARTIAL FRACTIONS • 1. PF1: Distinct Linear Factors • 2. PF2: Repeated Linear Factors • 3. PF3: Irreducible Quadratic • 4. Algebraic Long Division • 5. Improper Rational Functions • 2. DIFFERENTIATION • 1. From First Principles • 2. Product Rule • 3. Quotient Rule • 4. Trig. Terms • 5. ‘e’ • 6. Ln(x) • 7. Higher Derivatives • 8. Inverse Trig. Functions • 9. Implicit Diff 1 • 10. Implicit Diff 2: Equation of a Tangent • 11. Implicit Diff 3: Second Derivatives • 12. Logarithmic Diff • 13. Parametric Diff 1 • 14. Parametric Diff 2: Eqn of a Tangent • 15. Parametric Diff 3: Second Derivatives • 16. Parametric Diff 4: Objects in Motion • 3. INTEGRATION • 1. Standard Integrals • 2. ex • 3. Ln(x) • 4. Trig. Functions • 5. Int by Sub 1 • 6. Int by Sub 2: Fractions • 7. Int by Sub 3: Manipulation • 8. Int by Sub 4: Definite Integrals • 9. Int by Sub 5: √(a^2- x^2 ) • 10. Int by Sub 6: Substitution Not Given • 11. Inverse Trig. Functions • 12. Int using Partial Fractions 1 • 13. Int using Partial Fractions 2: Algebraic Long Division • 14. Int by Parts 1 • 15. Int by Parts 2: Harder Examples • 4. DIFFERENTIAL EQUATIONS • 1. Sep the Variables 1 • 2. Sep the Variables 2: Using Logs and Exponentials • 3. Sep the Variables 3: The Particular Solution • 4. Sep the Variables 4: Mathematical Models • 5. 1st Order Diff Eqn 1 • 6. 1st Order Diff Eqn 2: The Particular Solution • 7. 2nd Order Diff Eqn 1: Homogeneous • 8. 2nd Order Diff Eqn 2: Homogeneous – The Particular Solution • 9. 2nd Order Diff Eqn 3: Non-Homogeneous • 10. 2nd Order Diff Eqn 4: Non-Homogeneous – The Particular Solution • 11. 2nd Order Diff Eqn 5: Amended Particular Integral • 5. BINOMIAL THEOREM • 1. Factorials • 2. Binomial Coefficients • 3. Binomial Theorem / Expanding (x + y)n • 4. Finding Coefficients • • 6. SEQUENCES SERIES • 1. Sigma Notation • 2. The Sum of a Series with Partial Fractions • 3. The Method of Differences • 4. Summation Formulae • 5. Arithmetic Sequences • 6. The Sum of an Arithmetic Sequence • 7. Geometric Sequences • 8. The Sum of a Geometric Sequence • 9. Sum to Infinity • 10. MacLaurin Series 1 • 11. MacLaurin Series 2: Harder Examples • 7. PROOF BY INDUCTION • 1. Standard Examples • 2. Harder Examples • 8. FUNCTIONS GRAPHS • 1. Sketching Rational Functions • 2. Sketching Imp Rational Functions 1 • 3. Sketching Imp Rational Functions 2 (Stat Points Required) • 4. Sketching the Modulus Function • 5. Odd and Even Functions • 6. Stationary Points Using the Second Derivative • 7. Points of Inflection • 9. APPLICATIONS OF CALCULUS • 1. Displacement, Velocity and Acceleration • 2. Related Rates of Change 1 – Exp Defined Rates • 3. Related Rates of Change 2 – Imp Defined Rates • 4. The Area Between a Curve the y-axis • 5. Volumes of Solids of Revolution 1 X-axis • 6. Volumes of Solids of Revolution 2 Y-axis • 10. GAUSSIAN ELIMINATION MATRICES • 1. GE 1 • 2. GE 2 - Special Cases (Redundancy, Inconsistency Ill-Conditioning) • 3. An Introduction to Matrices • 4. Matrix Multiplication • 5. The Identity Matrix • 6. The Determinant of a Square Matrix • 7. Inverse of a Matrix • 8. Inverse of a 2x2 Matrix • 9. Inverse of a 3x3 Matrix • 10. Transformation Matrices • 11. Composition of Transformations • 11. VECTORS • 1. Eqn of a Line : Symmetric Form • 2. Eqn of a Line : Parametric Form • 3. The Intersection of 2 Lines • 4. The Angle Between 2 Lines • 5. The Eqn of a Plane • 6. The Vector Product • 7. The Angle Between Two Planes • 8. Line Plane Intersection Point • 9. The Angle Between a Line a Plane • 10. The Intersection of Planes • 11. The Shortest Dist from a Point to a Line • 12. The Shortest Dist from a Point to a Plane • 12. COMPLEX NUMBERS • 1. Introduction: Add/Subtract/Multiply/Divide • 2. Polynomial Equations • 3. Polar Form 1: Argand Diagram, Modulus Argument • 4. Polar Form 2: Multiplication Division • 5. De Moivre's Theorem • 6. Roots of Complex Numbers • 7. Trig. Identities • 8. Locus in the Complex Plane • 13. EUCLIDEAN ALGORITHM • 1. Number Bases (Division Algorithm) • 2. EA1: The Greatest Common Divisor • 3. EA2: Expressing the GCD in Terms of a and b • 14. PROOFS • 1. Disproof by Counter Example • 2. Direct Proof • 3. Negation Statements • 4. By Contradiction • 5. 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