Glossary

3.1 Definition of the Derivative

2 min read•august 7, 2024

Derivatives are the key to understanding how functions change. They measure the at any point, giving us a powerful tool for analyzing slopes, velocities, and other rates in various fields.

The is defined as the of a as the change approaches zero. This concept leads to important applications like finding tangent lines and calculating instantaneous rates of change in physics and economics.

Derivative Definition and Notation

Defining the Derivative

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  • Derivative measures the instantaneous rate of change of a function at a specific point
  • Defined as the limit of the difference quotient as the change in x approaches zero
    • Difference quotient: f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}, where hh represents a small change in xx
    • Limit definition of derivative: [f′(x)](https://www.fiveableKeyTerm:f′(x))=lim⁡h→0f(x+h)−f(x)h[f'(x)](https://www.fiveableKeyTerm:f'(x)) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}

Derivative Notation

  • Derivative of a function f(x)f(x) can be denoted in several ways:
    • f′(x)f'(x): prime notation
    • dydx\frac{dy}{dx}: Leibniz notation, emphasizing the derivative as a ratio of infinitesimal changes in yy and xx
    • ddxf(x)\frac{d}{dx}f(x): operator notation, treating ddx\frac{d}{dx} as an operator acting on the function f(x)f(x)
  • Choice of notation depends on context and personal preference

Geometric and Physical Interpretation

Tangent Line

  • Derivative at a point is equal to the to the function's graph at that point
    • is a straight line that touches the graph at a single point without crossing it
    • As hh approaches zero in the difference quotient, the (connecting two points on the graph) approaches the tangent line
  • Tangent line provides a linear approximation of the function near the point of tangency

Instantaneous Rate of Change

  • Derivative represents the instantaneous rate of change of a function at a specific point
    • (rate of change of position) in physics
    • (rate of change of total cost) in economics
  • Contrasts with , which is calculated over an interval
    • Average velocity (total displacement divided by total time)
    • Average cost (change in total cost divided by change in quantity)

Special Cases

One-Sided Derivatives

  • are used when a function's behavior differs on either side of a point
  • : f−′(x)=lim⁡h→0−f(x+h)−f(x)hf'_{-}(x) = \lim_{h \to 0^{-}} \frac{f(x+h)-f(x)}{h}, approaching xx from the left
    • Example: f(x)=∣x∣f(x) = |x| at x=0x=0, left-hand derivative is −1-1
  • : f+′(x)=lim⁡h→0+f(x+h)−f(x)hf'_{+}(x) = \lim_{h \to 0^{+}} \frac{f(x+h)-f(x)}{h}, approaching xx from the right
    • Example: f(x)=∣x∣f(x) = |x| at x=0x=0, right-hand derivative is 11
  • Function is differentiable at a point if and only if both one-sided derivatives exist and are equal
    • Example: f(x)=∣x∣f(x) = |x| is not differentiable at x=0x=0 because left and right-hand derivatives are different

Key Terms to Review (17)

Average rate of change: The average rate of change of a function over an interval is the ratio of the change in the function's values to the change in the input values, essentially representing how much the function's output changes per unit increase in the input. This concept is crucial as it provides insight into the behavior of functions over specific intervals, helping to bridge the understanding between linear and non-linear functions, and laying the groundwork for the definition of instantaneous rates of change through derivatives.
D/dx f(x): The expression d/dx f(x) represents the derivative of the function f(x) with respect to the variable x. It measures the rate at which the function changes as x changes, providing insight into the behavior of the function at any given point. This concept is fundamental in calculus, as it allows us to understand motion, growth, and change in various applications across mathematics and science.
Derivative: A derivative represents the rate at which a function changes at a certain point, essentially measuring how a function's output value changes as its input value changes. It provides valuable insights into the behavior of functions, such as identifying slopes of tangent lines and determining points of increasing or decreasing functions. Understanding derivatives is crucial for analyzing differentiability and continuity, as it highlights how these concepts relate to the smoothness and stability of a function's graph.
Difference quotient: The difference quotient is a mathematical expression that measures the average rate of change of a function over a specific interval. It is defined as $$\frac{f(x+h) - f(x)}{h}$$, where $$f$$ is a function, $$x$$ is a point in its domain, and $$h$$ is a small increment. This concept serves as the foundational step toward understanding derivatives, as it reflects how function values change relative to changes in their input.
Differentiable Function: A differentiable function is one that has a derivative at each point in its domain. This means that the function is smooth and continuous at those points, allowing us to determine the rate at which the function changes. The concept of differentiability connects closely with continuity and provides the foundation for understanding how functions behave, especially when applying various rules and theorems related to calculus.
Dy/dx: The term dy/dx represents the derivative of a function, indicating how the output value (y) changes with respect to a small change in the input value (x). This concept captures the rate of change of one variable relative to another and plays a critical role in understanding motion, growth, and various changes in real-world scenarios. It serves as a foundational idea in calculus, linking concepts such as slopes of tangent lines and instantaneous rates of change.
F'(x): The notation f'(x) represents the derivative of the function f(x) with respect to the variable x, indicating the rate at which the function's value changes as x varies. This concept is central to understanding how functions behave and provides insight into their continuity, differentiability, and the various rules for computing derivatives.
Instantaneous rate of change: The instantaneous rate of change refers to the rate at which a function is changing at a specific point, which can be understood as the slope of the tangent line to the curve at that point. This concept is essential for understanding how functions behave at precise moments and connects deeply with differentiability, continuity, and the derivative's interpretations.
Instantaneous velocity: Instantaneous velocity is defined as the velocity of an object at a specific moment in time. It represents the limit of the average velocity as the time interval approaches zero, giving a precise measure of how fast an object is moving and in which direction at that exact instant. This concept is crucial for understanding motion and is closely tied to derivatives, where it can be represented mathematically as the derivative of the position function with respect to time.
Left-hand derivative: The left-hand derivative is a specific type of derivative that evaluates the rate of change of a function as the input approaches a certain value from the left side. It focuses on the behavior of the function just before reaching that point, providing insight into its slope and continuity. Understanding the left-hand derivative is crucial for analyzing functions at points where they may not be differentiable or where behavior changes dramatically.
Limit: A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave at specific points, particularly where they may not be explicitly defined, such as discontinuities or asymptotes. Limits are essential for defining derivatives and integrals, providing the foundation for many advanced mathematical concepts.
Marginal Cost: Marginal cost refers to the additional cost incurred by producing one more unit of a good or service. It is a crucial concept in economics and calculus, as it helps businesses determine how much it costs to increase production and informs decision-making on pricing and output levels. Understanding marginal cost involves interpreting how costs change with respect to production levels, making it inherently tied to derivatives and rates of change.
One-Sided Derivatives: One-sided derivatives refer to the derivatives of a function calculated from only one side of a point. Specifically, they are classified as the left-hand derivative and the right-hand derivative, which provide insights into the behavior of the function as it approaches that point from either the left or the right. Understanding one-sided derivatives is crucial for analyzing functions at points where they may not be differentiable in the traditional sense, revealing potential discontinuities or sharp turns in their graphs.
Right-hand derivative: The right-hand derivative is a concept in calculus that represents the limit of the difference quotient of a function as it approaches a specific point from the right side. This means you look at values of the function that are slightly greater than the point in question. It is a crucial concept for understanding how functions behave near points of interest, especially where they might not be smooth or continuous.
Secant Line: A secant line is a straight line that intersects a curve at two or more points. This concept is crucial when studying the behavior of functions, as it helps in understanding how a function changes between those points. Secant lines provide a way to approximate the slope of the tangent line, which represents the instantaneous rate of change of the function at a particular point.
Slope of the tangent line: The slope of the tangent line at a specific point on a curve represents the instantaneous rate of change of the function at that point. This concept is crucial in understanding how functions behave and is directly related to the derivative, which quantifies how a function changes as its input varies. The slope indicates the steepness and direction of the curve at that particular point, providing vital information about the function's local behavior.
Tangent Line: A tangent line is a straight line that touches a curve at a single point without crossing it, representing the instantaneous rate of change of the curve at that point. This concept is deeply tied to the idea of differentiability, as a function must be differentiable at a point for a tangent line to exist there, which connects to continuity and basic differentiation principles.
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