5. Limits & Derivatives



5.1 Limits

  1. Definition & Notation

  2. Continuity

  3. Arithmetic Properties

  4. Limits of Lines & Asymptotes

  5. Squeeze Theorem

  6. Limits of Trig Functions

  7. Limits Related to Zero & Infinity


5.2 Mathematical Constants

  1. Proof of$\medspace\pi$ 🔧

  2. The Natural Number$\medspace e$ 🔧

  3. The Golden Ratio$\medspace\phi$

  4. The Plastic Ratio $\rho$


5.3 Differentiation

  1. Derivative Definition & Function

  2. Notation

  3. Tangent Line 🔧

  4. Differentiability 🔧

  5. First Derivative Use

  6. Second Derivative Use


5.4 Derivatives Common Rules

  1. Arithmetic Properties

  2. Product Rule

  3. Quotient Rule

  4. Power Rule

  5. Mean Value Theorem for Derivatives

  6. Implicit Differentiation (Chain Rule)

  7. Derivative of an Inverse Function

  8. L'Hôpital's Rule

  9. $d/dx\thinspace e^x$

  10. Derivative of Logarithmic Functions 🔧

  11. Derivative of Exponential Functions

  12. $d/dx\thinspace x^x$


5.5 Trigonometric Derivatives

  1. $d/d\theta\sin(\theta)$

  2. $d/d\theta\cos(\theta)$

  3. $d/d\theta\tan(\theta)$

  4. $d/d\theta\cot(\theta)$

  5. $d/d\theta\sec(\theta)$

  6. $d/d\theta\csc(\theta)$

  7. $d/dx\sin^{-1}(x)$

  8. $d/dx\cos^{-1}(x)$ 🔧

  9. $d/dx\tan^{-1}(x)$ 🔧

  10. $d/dx\cot^{-1}(x)$ 🔧

  11. $d/dx\sec^{-1}(x)$ 🔧

  12. $d/dx\csc^{-1}(x)$ 🔧


5.6 Complex Analysis & Trigonometry

  1. Complex Number System

  2. Analytic Function for $\sin(\theta)$

  3. Analytic Function for $\cos(\theta)$

  4. Analytic Function for $\sec(\theta)$

  5. Analytic Function for $\csc(\theta)$

  6. Analytic Function for $\tan(\theta)$

  7. Analytic Function for $\cot(\theta)$

  8. Analytic Function for $\sin^{-1}(x)$

  9. Analytic Function for $\cos^{-1}(x)$

  10. Analytic Function for $\csc^{-1}(x)$

  11. Analytic Function for $\sec^{-1}(x)$

  12. Analytic Function for $\tan^{-1}(x)$

  13. Analytic Function for $\cot^{-1}(x)$


5.7 Hyperbolic Trigonometry

  1. Unit Hyperbola Definitions

  2. Hyperbolic Common Identities

  3. Limits of Hyperbolic Functions

  4. Derivatives of Hyperbolic Functions

  5. Real Version of Euler's Formula

  6. Analytic Hyperbolic Functions

  7. Analytic Hyperbolic Inverses


5.8 Polar & Parametric Differentiation