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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  • Second derivatives
  • Second derivatives of implicit equations
  • Example
  • Example
  1. Differential Calculus

Higher Order Derivatives

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Last updated 6 years ago

  • First order derivative: f'(x) or dy/dx, also called "First Derivative"

  • Second order derivative: f''(x) or d²y/dx², also called "Second Derivative"

  • ...

Second derivatives

The second derivative of a function is simply the derivative of the function's derivative.

Notation: Leibniz's notation for second derivative is:

Second derivatives of implicit equations

Example

Solve:

Example

Solve:

  • The first derivative is:

  • The second derivative is:

  • Plug in y=1, and get:

▶️
▶ Back to previous note on Implicit Differentiation
▶ Practice at Khan academy: Second derivatives (implicit equations)
▶ Online calculator for 2nd Derivative of Implicit equations
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