Differential Calculus Overview

Differential Calculus

• Definition

  • Branch of calculus that deals with the rate of change of functions.
  • Primarily concerned with derivatives and their applications.

• Derivative

  • The derivative of a function ( f(x) ) at a point ( x ) is the limit of the average rate of change as the interval approaches zero: [ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} ]
  • Represents the slope of the tangent line to the curve at that point.

• Notation

  • ( f'(x) ) (Lagrange notation)
  • ( \frac{dy}{dx} ) (Leibniz notation)
  • ( Df(x) ) (Newton's notation)

• Rules of Differentiation

  • Power Rule: [ \frac{d}{dx}(x^n) = nx^{n-1} ]
  • Product Rule: [ \frac{d}{dx}(uv) = u'v + uv' ]
  • Quotient Rule: [ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2} ]
  • Chain Rule: [ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} ]

• Higher Order Derivatives

  • The second derivative ( f''(x) ) is the derivative of the derivative ( f'(x) ).
  • Higher derivatives can be denoted as ( f^{(n)}(x) ).

• Applications of Derivatives

  • Finding Tangents: Slope of the tangent line at a point.
  • Optimization: Finding local maxima and minima using the First and Second Derivative Tests.
  • Curve Sketching: Analyzing increasing/decreasing intervals and concavity.
  • Motion Analysis: Relating the derivative to velocity and acceleration in physics.

• Critical Points

  • Points where ( f'(x) = 0 ) or ( f'(x) ) is undefined.
  • Important for determining local maxima and minima.

• Concavity and Inflection Points

  • Concave Up: ( f''(x) > 0 ) indicates the function is curving upwards.
  • Concave Down: ( f''(x) < 0 ) indicates the function is curving downwards.
  • Inflection Point: Points where concavity changes, found where ( f''(x) = 0 ) or undefined.

• Special Functions

  • Trigonometric, logarithmic, and exponential functions have specific differentiation formulas.
    • Example:
      • ( \frac{d}{dx}(\sin x) = \cos x )
      • ( \frac{d}{dx}(e^x) = e^x )
      • ( \frac{d}{dx}(\ln x) = \frac{1}{x} )

These notes provide an overview of key concepts and techniques in differential calculus, essential for understanding more complex calculus topics and their applications.

Differential Calculus

• It's a branch of calculus focusing on the rate of change of functions.

• Derivatives are the primary focus, representing the instantaneous rate of change at a specific point.

Finding Derivatives

• The derivative of a function (f(x)) at (x) is found by taking the limit of the average rate of change over an interval approaching zero:

  [ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} ]

• This derivative represents the slope of the tangent line to the curve at that point.

Derivative Notations

• ( f'(x) ) is called Lagrange notation.

• ( \frac{dy}{dx} ) is known as Leibniz notation.

•  ( Df(x) ) is called Newton's notation.

Differentiation Rules

• Power Rule: For any power (n), ( \frac{d}{dx}(x^n) = nx^{n-1} ).

• Product Rule: For functions (u) and (v), ( \frac{d}{dx}(uv) = u'v + uv' ).

• Quotient Rule: For functions (u) and (v), ( \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2} ).

• Chain Rule: For a composite function (y = f(u)) where (u = g(x)), ( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} ).

Higher Order Derivatives

• The second derivative (f''(x)) is the derivative of the first derivative (f'(x)).

• Higher derivatives can be denoted as (f^{(n)}(x)).

Applications of Derivatives

• Finding Tangents: The derivative directly gives the slope of a tangent line to the function at any given point.

• Optimization: The first and second derivative tests help find local maxima and minima, crucial for optimization problems.

• Curve Sketching: Derivatives tell us where a function is increasing or decreasing, and its concavity (whether it curves upward or downward).

• Motion Analysis: Derivatives are used to define velocity (rate of change of position) and acceleration (rate of change of velocity) in physics.

Critical Points

• These occur when (f'(x) = 0) or (f'(x)) is undefined.
• They are pivotal for determining local maxima, minima, and inflection points.

Concavity and Inflection Points

• Concave Up: (f''(x) > 0) means the function is curving upwards.

• Concave Down: (f''(x) < 0) means the function is curving downwards.

• Inflection Point: When the concavity changes, found where (f''(x) = 0) or undefined.

Special Functions

• Trigonometric, logarithmic, and exponential functions have specific differentiation formulas.
• Some examples:
  • ( \frac{d}{dx}(\sin x) = \cos x )
  • ( \frac{d}{dx}(e^x) = e^x )
  • ( \frac{d}{dx}(\ln x) = \frac{1}{x} )

Knowing these concepts and rules is essential for navigating more complex calculus topics and their applications in real-world scenarios.

Differential Calculus

• Differential Calculus deals with the study of rates at which quantities change.
• Focuses on the concept of the derivative.
• Derivative represents the rate of change of a function concerning its variable.

Key Concepts

• Derivative is denoted as f'(x) or df/dx.

Notation

• Leibniz Notation: df/dx
• Lagrange Notation: f'(x)
• Newton Notation: ẋ (when time is the variable)

Basic Rules for Differentiation

• Power Rule: If f(x) = x^n, then f'(x) = n * x^(n-1).
• Product Rule: (uv)' = u'v + uv'
• Quotient Rule: (u/v)' = (u'v - uv') / v^2
• Chain Rule: If y = f(g(x)), then dy/dx = f'(g(x)) * g'(x)

Higher Order Derivatives

• Second Derivative: f''(x) or d²f/dx², indicates a function's concavity.
• n-th Derivative: f^(n)(x) or d^nf/dx^n, represents the derivative taken n times.

Applications of Derivatives

• Tangent Lines: The derivative at a point gives the slope of the tangent to the curve at that point.
• Optimization: Derivatives are used to find local maxima and minima by setting f'(x) = 0.
• Motion problems: Derivatives describe velocity and acceleration in physics.

Critical Points

• Points where the derivative is either zero or undefined.
• Determine local extrema through the First and Second Derivative Tests.

Continuity and Differentiability

• A function must be continuous at a point to be differentiable there.
• A differentiable function can have sharp turns or cusps where the derivative does not exist.

Differential Equations

• Equations that involve derivatives of functions are used to model real-world phenomena.

Limit Definition of Derivative

• f'(a) = lim (h → 0) [(f(a + h) - f(a)) / h]

Important Theorems

• Mean Value Theorem: If a function is continuous on [a, b] and differentiable on (a, b), then there's at least one point c in (a, b) where f'(c) = (f(b) - f(a)) / (b - a).
• Rolle's Theorem: If f(a) = f(b), then there's some c in (a, b) such that f'(c) = 0.