Calculus Derivatives Basics

Definition

• A derivative measures how a function changes as its input changes.
• It represents the rate of change or the slope of the function at a particular point.

Notation

• Common notations for derivatives include:
  • ( f'(x) ) or ( \frac{df}{dx} ) for a function ( f ).
  • ( Df ) for the derivative operator.

Basic Concepts

• Instantaneous Rate of Change: The derivative gives the slope of the tangent line to the curve of the function at a specific point.
• Tangent Line: The line that touches the function at a point without crossing it.

Rules of Differentiation

1. Power Rule:
   • ( \frac{d}{dx}(x^n) = nx^{n-1} )
2. Product Rule:
   • ( \frac{d}{dx}(uv) = u'v + uv' )
3. Quotient Rule:
   • ( \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2} )
4. Chain Rule:
   • ( \frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x) )

Higher-Order Derivatives

• The second derivative ( f''(x) ) is the derivative of the derivative, indicating the curvature of the function.
• Higher-order derivatives can be denoted as ( f^{(n)}(x) ).

Applications

• Finding Slopes: Determine the slope of curves at points.
• Optimization: Identify maximum and minimum values of functions using critical points.
• Motion: Derivatives represent velocity and acceleration in physics.

Special Functions

• Exponential Functions:
  • ( \frac{d}{dx}(e^x) = e^x )
• Trigonometric Functions:
  • ( \frac{d}{dx}(\sin x) = \cos x )
  • ( \frac{d}{dx}(\cos x) = -\sin x )

Implicit Differentiation

• Used when functions are defined implicitly rather than explicitly.
• Differentiate both sides of the equation with respect to ( x ) and solve for ( \frac{dy}{dx} ).

Summary

• Derivatives are foundational concepts in calculus that describe how functions behave.
• Mastery of differentiation rules and applications is essential for solving practical problems in mathematics and sciences.

Definition

• A derivative quantifies the change in a function's output in response to changes in its input.
• It signifies the rate of change or slope at specific points on a graph.

Notation

• Common derivative notations include:
  • ( f'(x) ) or ( \frac{df}{dx} ) to represent the derivative of a function ( f ).
  • ( Df ) for indicating the differentiation operator.

Basic Concepts

• Instantaneous Rate of Change: The derivative indicates the slope of the tangent line at any point on a function's curve.
• Tangent Line: A straight line that touches the function at one point without intersecting it.

Rules of Differentiation

• Power Rule:
  • The derivative of ( x^n ) is calculated as ( nx^{n-1} ).
• Product Rule:
  • For the product of two functions ( u ) and ( v ), the derivative is ( u'v + uv' ).
• Quotient Rule:
  • For the quotient of two functions ( u ) and ( v ), the derivative is ( \frac{u'v - uv'}{v^2} ).
• Chain Rule:
  • For composite functions, the derivative is given by ( f'(g(x)) \cdot g'(x) ).

Higher-Order Derivatives

• The second derivative, denoted as ( f''(x) ), measures the curvature of the function.
• Higher-order derivatives are expressed as ( f^{(n)}(x) ) for ( n )th derivatives.

Applications

• Finding Slopes: Derivatives help pinpoint the slope at specific points along curves.
• Optimization: Critical points identified through derivatives help find function maxima and minima.
• Motion: Derivatives represent physical concepts like velocity and acceleration.

Special Functions

• Exponential Functions: The derivative of ( e^x ) remains ( e^x ).
• Trigonometric Functions:
  • ( \frac{d}{dx}(\sin x) ) yields ( \cos x ).
  • ( \frac{d}{dx}(\cos x) ) results in ( -\sin x ).

Implicit Differentiation

• This technique allows differentiation when functions are defined in an implicit manner.
• To find ( \frac{dy}{dx} ), differentiate both sides of the equation with respect to ( x ) and solve accordingly.

Summary

• Derivatives are central to calculus, offering insights into function behavior.
• Proficiency in differentiation rules and real-world applications is critical for problem-solving across various mathematical and scientific domains.