Calculus Basics
  • Introduction
  • ▶️Limit & Continuity
    • Limit properties & Limits of Combined Functions
    • Limits at infinity
    • All types of discontinuities
  • ▶️Differential Calculus
    • Differentiability
    • Local linearity & Linear approximation
    • Basic Differential Rules
    • Chain Rule
    • Derivatives of Trig functions
    • Implicit differentiation
    • Higher Order Derivatives
    • Derivative of Inverse functions
    • Derivative of exponential functions
    • Existence Theorems
    • L'Hopital's Rule
    • Critical points
      • Extrema: Maxima & Minima
      • Concavity
      • Inflection Point
    • Second Derivative Test
    • Anti-derivative
    • Analyze Function Behaviors with Derivatives
    • Optimization
    • Applications of Derivatives
      • Motion problems
      • Planar motion
  • ▶️Integral Calculus
    • Definite Integrals
    • Antiderivatives
    • Fundamental Theorem of Calculus (FTC)
    • Basic Integral Rules
    • Calculate Integrals
    • Integration using Trig identities
    • Improper Integral
    • U-substitution → Chain Rule
    • Integrate by Parts → Product Rule
    • Partial fractions → Log Rule
    • Trig-substitutions → Trig Rule
    • Average Value of Functions
  • ▶️Differential Equations
    • Parametric Equations Differentiation
    • Separable Differential Equations
    • Specific antiderivatives
    • Polar Curve Functions (Differential Calc))
    • Logistic Growth Model
    • Slope Field
    • Euler's Method
  • ▶️Applications of definite integrals
  • ▶️Series (Calculus)
    • Infinite Seires
    • Infinite Geometric Series
    • Convergence Tests
      • nth Term Test
      • Integral Test
      • p-series Test
      • Comparison Test
      • Ratio Test
      • Root Test
      • Alternating Series Test
    • Absolute vs. Conditional Convergence
      • Error Estimation of Alternating Series
      • Error Estimation Theorem
      • Interval of Convergence
    • Power Series
      • Taylor Series
      • Maclaurin Series
      • Lagrange Error Bound
      • Finding Taylor series for a function
      • Function as a Geometric Series
      • Maclaurin Series of Common functions
      • Euler's Formula & Euler's Identity
  • Multivariable functions
    • Parametric Functions
    • Partial derivatives
    • Gradient
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  • Not differentiable situations
  • Vertical Tangent
  • Horizontal Tangent
  1. Differential Calculus

Differentiability

PreviousDifferential CalculusNextLocal linearity & Linear approximation

Last updated 6 years ago

"If the point of a function IS differentiable, then it MUST BE continuous at the point."

Example of NOT differentiable points:

You can see, if the point DOES NOT have limit, it's NOT DIFFERENTIABLE. In another word, the point is not CONTINUOUS, it's Jump Discontinuity, or Removable Discontinuity, or any type of discontinuities.

Not differentiable situations

  • Vertical Tangent (∞)

  • Not Continuous

  • Two sides' limits are different

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Vertical Tangent

We know that the Slope of Vertical Tangent is UNDEFINED, on the contrary: IT IS A VERTICAL TANGENT, IF:

  • The derivative dy/dx = undefined, or

  • The denominator of derivative's expression = 0.

Horizontal Tangent

It's a Horizontal Tangent, if:

  • dy/dx = 0.

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