Calculus Chapter 1: Derivatives from First Principles

Derivatives from First Principles: Limit Definition

What is a Derivative?

• A derivative represents the rate of change of a function with respect to one of its variables
• It measures how fast the output changes when one of the inputs changes

The Concept of Limits

• A limit represents the behavior of a function as the input (or x-value) approaches a specific point
• In the context of derivatives, limits help us understand how a function changes as the input changes

The Limit Definition of a Derivative

• The derivative of a function f(x) at a point x=a is denoted as f'(a) and is defined as:

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

• This definition states that the derivative is the limit of the ratio of the change in the output (f(a + h) - f(a)) to the change in the input (h) as h approaches 0

Interpretation of the Limit Definition

• The limit definition of a derivative can be interpreted as the instantaneous rate of change of the function at a point
• It measures the rate at which the output changes when the input changes by an infinitesimally small amount

Importance of the Limit Definition

• The limit definition of a derivative provides a rigorous mathematical framework for understanding rates of change
• It is used to develop many important concepts in calculus, including rules for differentiating functions and optimization techniques

Derivatives from First Principles: Limit Definition

What is a Derivative?

• A derivative represents the rate of change of a function with respect to one of its variables
• It measures how fast the output changes when one of the inputs changes

The Concept of Limits

• A limit represents the behavior of a function as the input (or x-value) approaches a specific point
• In the context of derivatives, limits help us understand how a function changes as the input changes

The Limit Definition of a Derivative

• The derivative of a function f(x) at a point x=a is denoted as f'(a) and is defined as: f'(a) = lim(h → 0) [f(a + h) - f(a)]/h
• This definition states that the derivative is the limit of the ratio of the change in the output to the change in the input as h approaches 0

Interpretation of the Limit Definition

• The limit definition of a derivative can be interpreted as the instantaneous rate of change of the function at a point
• It measures the rate at which the output changes when the input changes by an infinitesimally small amount

Importance of the Limit Definition

• The limit definition of a derivative provides a rigorous mathematical framework for understanding rates of change
• It is used to develop many important concepts in calculus, including rules for differentiating functions and optimization techniques