Partial derivatives, introduction

Khan Academy
11 May 201610:55
EducationalLearning
32 Likes 10 Comments

TLDRThis video script delves into the concept of partial derivatives for multi-variable functions, drawing parallels with ordinary derivatives. It explains the process of calculating partial derivatives by treating one variable as constant while differentiating with respect to the other. The script uses the function F(X, Y) = X^2 * Y + sin(Y) as an example to illustrate the computation of partial derivatives both at a specific point and as a general formula. It emphasizes the importance of understanding derivatives beyond the graphical representation, especially in higher dimensions, by focusing on the ratio of output change to input change.

Takeaways
  • 📚 The script introduces the concept of partial derivatives for multi-variable functions, emphasizing their similarity to ordinary derivatives.
  • 🔍 It explains the notation for ordinary derivatives using Leibniz notation, highlighting the interpretation of 'df/dx' as the change in output for a small change in input at a specific point.
  • 📈 The script uses a graphical approach to explain derivatives, showing how the slope of a tangent line on a graph represents the derivative at a point.
  • 📝 The analogy of 'nudging' the input variable to see the resulting change in the output is used to describe how derivatives are conceptualized.
  • 📉 The process of taking partial derivatives is described, where one treats all other variables as constants while differentiating with respect to one variable.
  • 📌 The script clarifies the notation used for partial derivatives, which involves a 'curl' at the top of the 'd' to indicate a partial derivative, such as ∂F/∂X.
  • 🤔 The reason behind the term 'partial' is explained, indicating that each partial derivative only provides a part of the full story of how the function changes.
  • 📑 The script provides a step-by-step example of calculating the partial derivatives of a given function F(X, Y) = X^2 * Y + sin(Y) at a specific point (1, 2).
  • 🔢 It demonstrates how to find both ∂F/∂X and ∂F/∂Y, treating Y and X as constants respectively, and then evaluates these at the given point.
  • 📘 The general formula for partial derivatives is explained, showing how to differentiate the function with respect to one variable while considering the other as constant without initially substituting a specific value.
  • 🔬 The script concludes by emphasizing the broader concept of derivatives beyond graphs and slopes, which is essential for understanding multi-variable calculus in higher dimensions.
Q & A

  • What is a multi-variable function?

    -A multi-variable function is a function that takes more than one input variable, such as F(X, Y), and returns a single scalar value as output.

  • How is a partial derivative similar to an ordinary derivative?

    -A partial derivative is similar to an ordinary derivative in that it measures the rate of change of a function with respect to one variable, treating the other variables as constants.

  • What is the Leibniz notation for derivatives?

    -The Leibniz notation for derivatives is a way to express the derivative of a function, often written as df/dx, which suggests the change in the function (df) due to a small change in the variable (dx).

  • How is the derivative of a function interpreted in terms of a graph?

    -In terms of a graph, the derivative of a function represents the slope of the tangent line to the curve at a particular point, which is the rate of change of the output for a small change in the input.

  • What does the 'D' with a curl at the top symbolize in mathematics?

    -The 'D' with a curl at the top is a symbol used to denote a partial derivative, emphasizing that the derivative is being taken with respect to one variable in a multi-variable function.

  • Why are partial derivatives called 'partial'?

    -Partial derivatives are called 'partial' because they provide only a part of the complete rate of change of a function, focusing on changes in one direction while holding other variables constant.

  • How do you compute the partial derivative of a function with respect to a specific variable?

    -To compute the partial derivative of a function with respect to a specific variable, you treat all other variables as constants and differentiate the function as you would with an ordinary derivative.

  • What is the difference between computing a partial derivative at a specific point and a general formula for a partial derivative?

    -Computing a partial derivative at a specific point involves substituting the values of the variables into the function and then differentiating. A general formula for a partial derivative, on the other hand, is an expression that can be used to find the derivative for any values of the variables.

  • How does the concept of 'nudging' the input help in understanding derivatives?

    -The concept of 'nudging' the input helps in understanding derivatives by visualizing a small change in the input variable and observing the resulting change in the output, which is then used to calculate the derivative as the ratio of these changes.

  • What is the significance of understanding derivatives beyond just graphs and slopes?

    -Understanding derivatives beyond just graphs and slopes is significant because it allows for the application of the concept to functions with higher dimensions or vector-valued functions where graphical representation is not feasible.

Outlines
00:00
📚 Introduction to Partial Derivatives

This paragraph introduces the concept of partial derivatives in the context of multi-variable functions. It begins by comparing partial derivatives to ordinary derivatives and aims to show their similarities. The speaker uses the function F(X, Y) = X^2 * Y + sin(Y) as an example and explains the notation for derivatives, including the Leibniz notation. The explanation includes a visual representation of how a small change in the input (nudge) affects the output, highlighting the concept of slope in the context of graphs. The paragraph sets the stage for understanding how to calculate partial derivatives by considering one variable at a time while treating the others as constants.

05:05
🔍 Calculating Partial Derivatives

The second paragraph delves into the actual process of calculating partial derivatives. It demonstrates how to find the partial derivative of the function F(X, Y) with respect to X and Y at a specific point (1, 2), treating the other variable as a constant. The speaker simplifies the function by substituting the constant value of the non-differentiated variable and then performs the derivative as an ordinary derivative. This results in specific values for the partial derivatives at the given point. The paragraph also discusses the general formula for partial derivatives, emphasizing the process of treating variables as constants during differentiation. The summary includes the general formulas obtained for ∂F/∂X and ∂F/∂Y, illustrating how these can be applied to any point (X, Y).

10:07
📘 Generalizing the Concept of Derivatives

The final paragraph emphasizes the broader concept of derivatives beyond the graphical representation of slopes. It explains that while graphs and slopes are useful for understanding derivatives in two dimensions, they are not applicable in higher dimensions or for vector-valued functions. The paragraph reinforces the idea of 'nudging' the input in a certain direction to see how it affects the output, and then taking the ratio of the output change to the input change as a more general approach to understanding derivatives. This approach is highlighted as essential for further studies in multi-variable calculus.

Mindmap
Keywords
💡Multi-variable function
A multi-variable function is a mathematical function that accepts more than one input variable. In the context of the video, the function F of XY is an example where it takes two variables, X and Y, as inputs. This concept is central to the video's theme of discussing partial derivatives, which are used to measure how changes in one variable affect the function's output while holding other variables constant.
💡Partial derivative
A partial derivative is a derivative of a function with respect to one of its variables, treating the other variables as constants. The video explains that partial derivatives are similar to ordinary derivatives but are used in multi-variable functions to understand the rate of change of the function with respect to one variable at a time. The script uses the function F(X, Y) = X^2 * Y + sin(Y) to demonstrate how to calculate partial derivatives with respect to X and Y.
💡Leibniz notation
Leibniz notation is a way of expressing derivatives, often written as df/dx, which represents the derivative of a function f with respect to variable x. The video uses this notation to explain the concept of ordinary derivatives and contrasts it with the notation for partial derivatives, emphasizing the visual and intuitive understanding of derivatives as slopes on a graph.
💡Graph
In the video, a graph is used as a visual representation of a function's behavior. The script explains how the graph of a function like x squared has a parabolic shape and how the derivative can be interpreted as the slope of the tangent line at a particular point on the graph. This visual aid helps in understanding the concept of derivatives and their geometric interpretation.
💡Slope
Slope is a concept from geometry that describes the steepness of a line, and in calculus, it is used to describe the rate of change of a function at a particular point. The video script uses the term 'slope' to explain the derivative as the ratio of the change in the output (rise) to the change in the input (run), which is a fundamental concept in understanding derivatives.
💡Input space
Input space refers to the set of all possible input values for a function. In the context of the video, when discussing multi-variable functions, the input space is visualized as an XY plane where each point on the plane represents a pair of input values (X, Y). This concept is essential for understanding how partial derivatives can be used to analyze the function's behavior in multi-dimensional spaces.
💡Output
Output in the context of the video refers to the result produced by a function given a set of input values. The script discusses how the output changes in response to a 'nudge' or small change in the input values, which is the fundamental idea behind calculating derivatives and partial derivatives.
💡Constant
In the script, a constant is a value that does not change during the differentiation process. When calculating a partial derivative, all other variables are treated as constants. For example, when taking the partial derivative of F with respect to X, Y is considered a constant, and vice versa. This allows the focus to be solely on how changes in one variable affect the output.
💡Ordinary derivative
An ordinary derivative is the derivative of a function with respect to a single variable. The video script contrasts ordinary derivatives with partial derivatives, explaining that the process of finding an ordinary derivative is similar to finding a partial derivative, but without the need to consider other variables as constants.
💡General formula
A general formula in the context of the video refers to a formula that can be applied to any point in the input space to find the partial derivative. The script explains how to derive such formulas for partial derivatives with respect to X and Y, which can then be used to find the rate of change at any point (X, Y) in the input space.
💡Vector-valued functions
Vector-valued functions are a type of function that outputs a vector rather than a single scalar value. The video script mentions vector-valued functions to illustrate that the concept of derivatives extends beyond functions with scalar outputs, and the idea of 'nudging' the input in a direction becomes more abstract in such cases.
Highlights

Introduction to multi-variable functions and the concept of partial derivatives.

Comparison between partial derivatives and ordinary derivatives, highlighting their similarities.

Explanation of the Leibniz notation for derivatives and its interpretative advantages.

Graphical representation of the derivative as the slope of a function's tangent line.

Interpretation of 'dx' as a small nudge in the x-direction and its impact on the function's output.

Illustration of how to visualize multi-variable functions in the xy-plane.

Demonstration of how to calculate the partial derivative with respect to x by treating y as a constant.

Calculation of the partial derivative with respect to y, emphasizing the constant treatment of x.

Introduction of the 'partial' notation and its significance in multi-variable calculus.

Explanation of why partial derivatives are termed 'partial' due to their focus on one variable at a time.

Step-by-step calculation of the partial derivative of F with respect to x at a specific point (1,2).

Step-by-step calculation of the partial derivative of F with respect to y at the same point (1,2).

General formula derivation for the partial derivative with respect to x, considering y as a constant.

General formula derivation for the partial derivative with respect to y, considering x as a constant.

Discussion on the importance of understanding partial derivatives beyond graphical interpretations.

Emphasis on the general approach to understanding derivatives through input and output nudges and their ratios.

Teaser for the next video, which will explore the meaning of partial derivatives in terms of graphs and slopes.

Transcripts

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Partial DerivativesMulti-variableCalculusDerivative NotationLeibniz NotationGraph InterpretationInput InfluenceFunction AnalysisEducational ContentMathematics Tutorial