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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On this page
  • Derivative notations
  • Lagrange's notation
  • Leibniz's notation
  • How to understand dy/dx
  • Tangent line & Secant line
  • Secant line
  • Example

Differential Calculus

PreviousAll types of discontinuitiesNextDifferentiability

Last updated 6 years ago

Simply saying, it's just the SLOPE of ONE POINT of a graph (line or curves or anything).

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A Derivative, is the Instantaneous Rate of Change, which's related to the tangent line of a point, instead of a secant line to calculate the Average rate of change.

“Derivatives are the result of performing a differentiation process upon a function or an expression. ”

Derivative notations

Lagrange's notation

In Lagrange's notation, the derivative of f(x) expressed as f'(x), reads as f prime of x.

Leibniz's notation

For memorizing, just see d as Δ, reads Delta, means change. So dy/dx means Δy/Δx. Or it can be represent as df / dx or d/dx · f(x), whatever.

How to understand dy/dx

This is a review from "the future", which means while studying Calculus, you have to come back constantly to review what the dy/dx means. ---- It's just so confusing. Without fully understanding the dy/dx, you will be lost at topics like Differentiate Implicit functions, Related Rates, Differential Equations and such.

Tangent line & Secant line

  • The secant line is drawn to connect TWO POINTS, and gets us the Average Rate of Change between two points.

  • The Tangent line is drawn through ONE POINT, and gets us the Rage of change at the exact moment.

As for the secant line, its interval gets smaller and smaller and APPROACHING to 0 distance, it actually is a process of calculating limits approaching 0, which will get us the tangent line, that been said, is the whole business we're talking about: the Derivative, the Instantaneous Rate of Change.

Secant line

Example

  • What it's asking is the Slope of its secant line:

  • which could be applied with this simple formula:

  • the result is ( f(3)-f(1) )/ (3-1) = -1/12

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In this form, we write dx instead of Δx heads towards 0. And the derivative of is commonly written as:

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secant to tangent animation

Solve:

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Refer to Khan academy article: Derivative notation review.
Refer to Khan academy from Differential Equation section: Addressing treating differentials algebraically
Refer to Mathsisfun: Introduction to Derivatives
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