Derivatives as Rate of Change

Introduction

The derivative of a function is a fundamental concept in calculus, representing the rate at which a quantity changes. In physical contexts, derivatives are used to describe velocity, acceleration, and other rates of change.

• Position Function: If s(t) denotes the position of an object at time t, then the derivative s'(t) gives the instantaneous velocity.

• Average Velocity: The average velocity over the interval [a, b] is given by:

• Instantaneous Velocity: The instantaneous velocity at time t = a is the limit of the average velocity as the interval shrinks:

• Acceleration: The acceleration is the derivative of velocity, or the second derivative of position:

• Example: Given , find velocity and acceleration.

Velocity: Acceleration: Initial velocity: Position at :

Rules of Differentiation

Basic Derivative Rules

Several standard rules are used to compute derivatives efficiently:

• Constant Rule: for any constant

• Power Rule:

• Exponential Rule:

• Logarithmic Rule:

• Trigonometric Rules:

• Sum Rule:

• Difference Rule:

• Product Rule:

• Quotient Rule:

Chain Rule

Introduction

The Chain Rule is a fundamental technique for differentiating composite functions. If a function is composed of two or more functions, the chain rule allows us to find its derivative efficiently.

• Statement: If , then:

• Notation: Let , then and

• Example 1:

Let , so

• Example 2:

Let , so

• Example 3:

Let , so

• Example 4:

Let , so

Applications and Importance

• The chain rule is essential for differentiating functions where one variable depends on another, such as in physics, engineering, and economics.

• It is used extensively in implicit differentiation and in finding derivatives of complicated expressions.

Summary Table: Derivative Rules

Additional info: Some context and examples were inferred and expanded for clarity and completeness.