Formal definition of partial derivatives

Khan Academy
11 May 201607:57
EducationalLearning
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TLDRThis video script offers a detailed exploration of partial derivatives, starting with a one-dimensional analogy to the ordinary derivative. It explains the concept of a 'nudge' in the input value and how this affects the output, formalizing the idea with the limit as the nudge approaches zero. The script then transitions to multi-variable functions, illustrating how to calculate partial derivatives with respect to different variables, emphasizing the importance of considering the limit for any specific size of the nudge. The explanation aims to clarify the formal definition of partial derivatives and their significance in calculus.

Takeaways
  • 📚 The script discusses the concept of partial derivatives, which is a fundamental aspect of calculus for functions with multiple variables.
  • 🔍 It begins with an analogy to the one-dimensional derivative, emphasizing the idea of a 'nudge' in the input value and its effect on the output.
  • 📈 The formal definition of a derivative is introduced, highlighting the limit process as the 'nudge' (denoted as 'H') approaches zero.
  • 📉 The script explains the notation and process of calculating the partial derivative with respect to one variable while keeping others constant.
  • 📝 The importance of the limit in calculus is emphasized, as it provides a rigorous framework for understanding infinitesimally small changes.
  • 📐 The concept is then extended to multi-variable functions, illustrating how to compute partial derivatives in higher-dimensional spaces.
  • 📊 The script uses visual aids like graphs and sketches to help understand the impact of a small change in one variable on the function's output.
  • 🔢 The formal definition of a partial derivative involves taking the limit of the ratio of the change in the function to the change in the variable as the latter approaches zero.
  • 📌 The script differentiates between partial derivatives with respect to different variables, such as X and Y, in a multi-variable function.
  • 📘 It also touches on the historical notation used in calculus, attributing the notation for derivatives to Leibniz, though it's noted that the origin of 'partials' is less certain.
  • 🔑 The takeaways from the script are intended to provide a clear understanding of partial derivatives, their formal definition, and their significance in multi-variable calculus.
Q & A
  • What is a partial derivative?

    -A partial derivative is a derivative of a function with multiple variables that measures how the function changes with respect to one variable while keeping the other variables constant.

  • Why is it called a 'partial' derivative?

    -It is called 'partial' because it considers the rate of change of a function with respect to one variable while holding all other variables constant, thus only taking a part of the full derivative.

  • How is the concept of a partial derivative related to the ordinary derivative?

    -The concept of a partial derivative is analogous to the ordinary derivative, but it applies to functions with multiple variables instead of just one, focusing on the change with respect to one variable at a time.

  • What does the notation ∂f/∂x represent?

    -The notation ∂f/∂x represents the partial derivative of the function f with respect to the variable x, indicating the rate at which f changes as x changes, with other variables held constant.

  • What is the role of 'H' in the definition of a partial derivative?

    -In the definition of a partial derivative, 'H' is used to represent a small increment or 'nudge' in the variable with respect to which the derivative is being taken, and the limit as H approaches zero is considered.

  • Why is the limit as H approaches zero important in the definition of a derivative?

    -The limit as H approaches zero is important because it formalizes the idea of an 'infinitesimal' change, which is the fundamental concept of a derivative, representing the instantaneous rate of change at a point.

  • How does the process of finding a partial derivative differ from finding an ordinary derivative?

    -The process of finding a partial derivative involves differentiating with respect to one variable while treating all other variables as constants, whereas an ordinary derivative involves differentiating a single-variable function.

  • What is the significance of evaluating the function at a specific point when finding a partial derivative?

    -Evaluating the function at a specific point is significant because it allows for the calculation of the change in the function's value due to a small increment in one of the variables, which is essential for determining the partial derivative at that point.

  • Can the concept of a partial derivative be extended to functions with more than two variables?

    -Yes, the concept of a partial derivative can be extended to functions with more than two variables. Each partial derivative would be taken with respect to one variable at a time, holding all other variables constant.

  • What is the relationship between the partial derivative and the graph of a multi-variable function?

    -The partial derivative describes the slope of the tangent plane to the graph of a multi-variable function along the direction of the variable with respect to which the derivative is taken, while holding other variables constant.

  • How does the formal definition of a partial derivative help in understanding the concept of a directional derivative?

    -The formal definition of a partial derivative provides a foundation for understanding the directional derivative, which involves the rate of change of a function in a specific direction in the multi-variable space, by considering the limit of the ratio of the change in function value to the change in the input along that direction.

Outlines
00:00
📚 Introduction to Partial Derivatives

The script begins by discussing the concept of partial derivatives, which are derivatives of functions with multiple variables. It draws an analogy with the ordinary derivative of a single-variable function, such as F(x) = x^2, to explain the process of taking a derivative. The concept of a 'nudge' in the input value is introduced, represented by the variable H, to illustrate the change in the function's output (DF) as the input (X) is slightly altered. The formal definition of a partial derivative is then outlined, emphasizing the limit as H approaches zero, which is central to calculus. The explanation includes a visual representation of the input space and output space, and how a small change in one of the input variables (X or Y) affects the output, leading to the formal definition of the partial derivative with respect to X.

05:01
🔍 Deep Dive into the Formal Definition of Partial Derivatives

This paragraph delves deeper into the formal definition of partial derivatives, focusing on the process of taking the limit as H approaches zero. The explanation clarifies that the size of the nudge (H) is not fixed; instead, the focus is on how the ratio of the change in the function's output to the change in the input approaches a certain value as H diminishes. The script then provides a practical example of writing out the partial derivative with respect to Y, which involves nudging in the Y direction while keeping the X value constant. The process involves evaluating the function at the original point and at the new point where Y has been incremented by H, and then taking the limit of the ratio of these differences to H as H approaches zero. The summary also touches on the notation used in partial derivatives, possibly originated by Leibniz, and hints at more complex multi-variable derivatives such as the directional derivative, setting the stage for further exploration in subsequent content.

Mindmap
Keywords
💡Partial Derivative
A partial derivative is a derivative of a function with respect to one of its variables, while the other variables are considered as constants. In the video, the concept is introduced as a formal definition for functions with multi-variable inputs, such as F(X, Y). The script uses the analogy of a one-dimensional derivative to explain the concept, emphasizing the limit process as the 'nudge' (represented by H) goes to zero, which is central to the calculus involved in partial derivatives.
💡Multi-variable Function
A multi-variable function is a mathematical function that has more than one input variable. In the context of the video, the focus is on functions like F(X, Y) where the partial derivatives are taken with respect to each variable individually. The script explains how to compute and interpret these derivatives, highlighting the importance of considering each variable's influence on the function's output.
💡Derivative
The derivative of a function measures the rate at which the function's output changes with respect to its input. The video script discusses the ordinary derivative as a one-variable function and then extends this concept to partial derivatives for multi-variable functions. The derivative is foundational to understanding the rate of change and is a key concept in calculus.
💡Limit
In calculus, the limit is the value that a function or sequence 'approaches' as the input or index approaches some value. The script explains the limit as H (representing a small change in the input) goes to zero, which is essential for defining the derivative and partial derivative. The limit process is crucial for understanding the instantaneous rate of change.
💡Variable Input
Variable input refers to the input values that a function takes. The video script discusses functions with multiple variable inputs, such as X and Y, and how each input affects the function's output. Understanding variable inputs is essential for computing partial derivatives, as each input is considered individually.
💡Ordinary Derivative
An ordinary derivative, also known as a single-variable derivative, is the rate of change of a function with respect to a single variable. The script uses the ordinary derivative as an analogy to introduce the concept of partial derivatives, emphasizing the idea of a 'nudge' in the input value and the resulting change in the function's output.
💡Nudge
In the script, 'nudge' is used to describe a small change in the input value of a function. It is a conceptual tool to help understand how the output of a function changes in response to an infinitesimal change in its input. The term is used to illustrate the process of taking a derivative or partial derivative.
💡Output Space
Output space refers to the set of all possible output values of a function. In the video, the script describes the function mapping the input space (like the X-Y plane) to the real number line, which represents the output space. Understanding output space is important for visualizing how changes in input affect the function's result.
💡Input Space
Input space is the set of all possible input values for a function. The script mentions the input space in the context of multi-variable functions, where the input space could be a high-dimensional space. The concept is crucial for understanding how partial derivatives are computed by considering changes in one dimension at a time.
💡Leibniz's Notation
Leibniz's notation is a mathematical notation used to represent derivatives, typically written as Df/Dx for the derivative of a function f with respect to x. The script references this notation when discussing partial derivatives, explaining how it conveys the idea of a change in the function (Df) over a change in the input (Dx).
💡Directional Derivative
A directional derivative is a generalization of the partial derivative that measures the rate of change of a function in a particular direction in space. While the script does not delve deeply into this concept, it is mentioned as an example of a more advanced notion of multi-variable derivatives, suggesting a further exploration of the topic.
Highlights

Introduction to the formal definition of partial derivatives.

Partial derivatives apply to functions with multi-variable inputs, such as X and Y.

Analogous to the one-dimensional derivative, the partial derivative is defined with respect to one variable at a time.

The concept of a 'nudge' in the input value is introduced to explain the derivative.

The formal definition of a derivative involves the limit as the nudge approaches zero.

The use of variable H to represent a small change in the input value.

Explanation of the partial derivative in the context of a multi-variable function.

Visualization of the input space as an XY plane or higher-dimensional space.

The partial derivative is calculated by considering a small change in one variable while keeping others constant.

The importance of taking the limit as H approaches zero in the definition of a partial derivative.

The formal definition of the partial derivative with respect to X, emphasizing the limit process.

Illustration of the partial derivative with respect to Y, highlighting the change in the Y direction.

The notation used in the formal definition of partial derivatives and its significance.

The connection between the formal definition of partial derivatives and the Leibniz notation.

The relevance of understanding partial derivatives for introducing more complex multi-variable derivatives.

The importance of rigor in the definition of partial derivatives for mathematical clarity.

A summary of the key points in understanding partial derivatives, preparing for further exploration in calculus.

Transcripts
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