Partial derivatives 2 | Multivariable Calculus | Khan Academy

Khan Academy

9 Aug 200810:26

EducationalLearning

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TLDRThe video script discusses the concept of partial derivatives in calculus, focusing on how changes in one variable affect the function while keeping others constant. It illustrates this with examples, including a symmetric function where partial derivatives with respect to x and y yield similar results. The script also demonstrates how to calculate partial derivatives for more complex functions with multiple variables, emphasizing the importance of identifying constants and variables in the process. The visualization of the function's surface and tangent planes highlights the practical application of partial derivatives.

Takeaways

📝 The concept of partial derivatives is introduced, which measures how a function changes with respect to one variable while keeping the other(s) constant.
🔺 The partial derivative of z with respect to y is calculated, demonstrating the treatment of x as a constant when finding the derivative of the term x squared.
📌 The derivative of a constant with respect to a variable is zero, which simplifies the process of finding partial derivatives.
🔄 Symmetry in the function f(x, y) leads to symmetrical partial derivatives; in this case, ∂z/∂x = 2x + y and ∂z/∂y = x + 2y.
📍 A specific point (x=0.3, y=0.3) is chosen to evaluate the function and its partial derivatives, resulting in z=0.27 and slopes of 0.9 in both x and y directions.
💠 Visualization of the function's surface and tangent planes is discussed, highlighting the concept of tangent lines and planes at a given point on the surface.
🌟 The importance of understanding the directions in the xy plane for finding tangent lines is emphasized, as there are infinite possibilities based on the chosen direction.
🎥 The script includes a walkthrough of graphing the function and its partial derivatives, providing a clear visual representation of the mathematical concepts.
🧩 The process of calculating partial derivatives for more complex functions with multiple variables is demonstrated, emphasizing the need to identify what variables are constant and which are changing.
🔢 Examples with varying numbers of dimensions are provided to illustrate the generalization of partial derivatives, showing how to handle different notations and variable interactions.

Q & A

What is the concept of a partial derivative?
-A partial derivative is a derivative of a function of multiple variables with respect to one variable, while treating all the other variables as constants.
How is the partial derivative of z with respect to y calculated in the given function?
-The partial derivative of z with respect to y is calculated by differentiating the function 'z = x^2 + xy + y^2' with respect to y, treating x as a constant. The result is ∂z/∂y = x + 2y.
What is the significance of treating x as a constant when finding the partial derivative with respect to y?
-Treating x as a constant simplifies the process of differentiation with respect to y. It allows us to ignore the term 'x^2' when differentiating, as the derivative of a constant is zero.
What is the partial derivative of z with respect to x at the point (x=0.3, y=0.3)?
-The partial derivative of z with respect to x at the point (x=0.3, y=0.3) is ∂z/∂x = 2x + y. Substituting x = 0.3 and y = 0.3, we get ∂z/∂x = 2(0.3) + 0.3 = 0.9.
How does the symmetry of the equation affect the partial derivatives?
-The symmetry of the equation 'z = x^2 + xy + y^2' means that the partial derivatives with respect to x and y are similar in form. Both are linear combinations of x and y, reflecting the symmetry in the variables' roles within the function.
What is the geometric interpretation of the partial derivatives at a specific point?
-The partial derivatives at a specific point represent the slopes of the tangent lines to the surface in the x and y directions, respectively. These slopes indicate the rate of change of the function in those directions at that point.
How can you visualize the partial derivatives on a graph?
-You can visualize the partial derivatives by plotting the surface and drawing tangent lines at a given point. The slopes of these tangent lines correspond to the partial derivatives, and their combination can help define a tangent plane at that point.
What is the partial derivative of the function f(x, y) = x * sin(x) * cos(y) with respect to x?
-The partial derivative of f(x, y) = x * sin(x) * cos(y) with respect to x is ∂f/∂x = sin(x) * cos(y) + x * cos(x) * cos(y), treating y as a constant.
What is the partial derivative of the function f(x, y) = x * sin(x) * cos(y) with respect to y?
-The partial derivative of f(x, y) = x * sin(x) * cos(y) with respect to y is ∂f/∂y = -x * sin(x) * sin(y), treating x as a constant.
How do you calculate the partial derivative of a composite function such as a^2 * b^3 * c^(1/2) with respect to a?
-The partial derivative of a composite function a^2 * b^3 * c^(1/2) with respect to a is ∂(a^2 * b^3 * c^(1/2))/∂a = 2a * b^3 * c^(1/2), treating b and c as constants.
How do you calculate the partial derivative of a composite function such as a^2 * b^3 * c^(1/2) with respect to b?
-The partial derivative of a composite function a^2 * b^3 * c^(1/2) with respect to b is ∂(a^2 * b^3 * c^(1/2))/∂b = 3a^2 * b^2 * c^(1/2), treating a and c as constants.
How do you calculate the partial derivative of a composite function such as a^2 * b^3 * c^(1/2) with respect to c?
-The partial derivative of a composite function a^2 * b^3 * c^(1/2) with respect to c is ∂(a^2 * b^3 * c^(1/2))/∂c = a^2 * b^3 * (1/2) * c^(-1/2), treating a and b as constants.

Outlines

00:00

📚 Introduction to Partial Derivatives

This paragraph introduces the concept of partial derivatives, specifically focusing on the partial derivative of a function with respect to one variable while treating the other variables as constants. The explanation begins with a visual representation of the function's surface and the concept of tangent lines and planes. The main points include the calculation of the partial derivative of z with respect to y, highlighting the symmetry in the process and the resulting expressions for the partial derivatives with respect to both x and y. The paragraph also discusses the selection of a specific point (x=0.3, y=0.3) and the computation of the corresponding z value and partial derivatives, emphasizing the symmetry and the slope of the tangent lines in both x and y directions.

05:01

🌟 Exploring Tangent Planes and Lines

This section delves deeper into the implications of partial derivatives by discussing how two lines can define a plane, and within a tangent plane, there are an infinite number of tangent lines. The focus is on understanding the mathematics behind partial derivatives and their applications in various dimensions. The explanation includes a new function defined by x sine of x and cosine of y, and the process of taking partial derivatives with respect to both x and y is demonstrated. The summary emphasizes the importance of recognizing which variables are constants and which are not, and how this affects the calculation of partial derivatives.

10:02

📝 Advanced Partial Derivative Examples

The final paragraph presents more complex examples of partial derivatives involving multiple variables. The examples include a function composed of variables a, b, and c, and the process of calculating partial derivatives with respect to each of these variables is detailed. The explanation covers the treatment of other variables as constants and the application of algebraic rules to simplify the resulting expressions. The paragraph aims to reinforce the understanding of partial derivatives by showcasing different notations and mathematical operations, emphasizing the need to keep track of what variables are considered constants in the process.

Mindmap

Keywords

💡partial derivative

A partial derivative is a mathematical concept that represents the rate at which a function changes with respect to one variable, while keeping all other variables constant. In the video, the partial derivative is used to determine the slope of a surface at a specific point, which can be visualized as the tangent to the surface at that point. For instance, the video discusses calculating the partial derivative of z with respect to x and y, resulting in ∂z/∂x = 2x + y and ∂z/∂y = x + 2y, highlighting the symmetry of the function.

💡constant

In the context of the video, a constant refers to a value that does not change when the function is being differentiated with respect to a particular variable. The derivative of a constant with respect to any variable is zero. This concept is crucial in partial differentiation because it simplifies the process by allowing us to ignore terms that do not contribute to the rate of change of the function concerning the variable of interest.

💡symmetry

Symmetry in the context of the video refers to the balanced and mirror-like relationship between different parts of a mathematical function. The symmetry of a function can make the partial derivatives with respect to different variables more manageable to compute and understand. In the video, the function z = x^2 + xy + y^2 is noted to be symmetric, as the x and y terms have similar roles, leading to the same value for the partial derivatives with respect to x and y at the point (0.3, 0.3).

💡tangent line

A tangent line is a straight line that touches a curve at a single point without crossing it. In the context of the video, the tangent line represents the local behavior of a surface at a given point. It is used to visualize the direction and rate of change of a function at that point. The slope of the tangent line corresponds to the partial derivative of the function at that point.

💡tangent plane

A tangent plane is a flat surface that touches a three-dimensional shape, like a curve or a surface, at a single point without crossing it. In the video, the concept of a tangent plane is used to describe the local geometry of a surface at a given point. It is formed by the intersection of the surface with a plane that contains the tangent lines at that point. The tangent plane provides a two-dimensional approximation of the surface near the point of tangency.

💡graphing

Graphing in the context of the video refers to the visual representation of mathematical functions and their properties. It involves using graphical tools to plot points, create lines, and visualize surfaces to better understand the behavior of functions and their derivatives. The video uses graphing to illustrate the concepts of partial derivatives, tangent lines, and tangent planes, providing a visual aid to the mathematical concepts discussed.

💡slope

Slope is a measure of the steepness of a line, indicating the rate of change of the y-values with respect to the x-values. In the context of the video, the slope is used to describe the rate at which the function changes in the x or y direction at a particular point. The slope is equivalent to the value of the partial derivative at that point.

💡derivative

A derivative is a fundamental concept in calculus that represents the rate of change or the slope of a function at a particular point. It is used to analyze the behavior of functions, such as their increases, decreases, and points of inflection. In the video, the derivative is specifically the partial derivative, which considers the change of the function with respect to one variable while holding others constant.

💡product rule

The product rule is a fundamental calculus formula used to find the derivative of the product of two functions. It states that the derivative of a product is the derivative of the first function times the second function plus the first function times the derivative of the second function. In the video, the product rule is applied when differentiating the term xy in the function z = x^2 + xy + y^2.

💡visualization

Visualization in the context of the video refers to the graphical representation and interpretation of mathematical concepts, such as partial derivatives, tangent lines, and tangent planes. It is a crucial tool for understanding the behavior of functions and their geometric implications in a three-dimensional space.

💡differentiation

Differentiation is the process of finding the derivative of a function, which describes the rate at which the function changes. It is a core concept in calculus and is used to analyze various aspects of functions, such as their maxima, minima, and inflection points. In the video, differentiation is applied to find partial derivatives with respect to different variables, which helps in understanding the local behavior of the function.

Highlights

Introduction to partial derivatives and their calculation.

Explanation of how to treat variables as constants when finding partial derivatives.

Derivation of the partial derivative of z with respect to y, highlighting the symmetry in the process.

Illustration of the symmetry in the function f(x, y) = x^2 + xy + y^2 through its partial derivatives.

Selection of a specific point (x=0.3, y=0.3) to evaluate the function and its partial derivatives.

Calculation of the value of z at the chosen point and verification of the partial derivatives at that point.

Visualization of the function's surface and the concept of tangent planes and lines.

Demonstration of how to graph the function and its tangent lines for a better understanding of partial derivatives.

Exploration of the concept of infinite tangent lines within a tangent plane at a given point on the function's surface.

Transition to additional partial derivative problems to familiarize with the mathematical process.

Solution of a multi-dimensional partial derivative problem involving trigonometric functions.

Explanation of how to handle constants and variables when calculating partial derivatives with multiple variables.

Another example of partial derivatives involving a polynomial expression and its simplification.

Demonstration of the importance of recognizing what variables are constant and what are changing during partial differentiation.

Conclusion and预告 of the next video for further exploration of partial derivatives.

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Takeaways

Q & A

What is the concept of a partial derivative?

How is the partial derivative of z with respect to y calculated in the given function?

What is the significance of treating x as a constant when finding the partial derivative with respect to y?

What is the partial derivative of z with respect to x at the point (x=0.3, y=0.3)?

How does the symmetry of the equation affect the partial derivatives?

What is the geometric interpretation of the partial derivatives at a specific point?

How can you visualize the partial derivatives on a graph?

What is the partial derivative of the function f(x, y) = x * sin(x) * cos(y) with respect to x?

What is the partial derivative of the function f(x, y) = x * sin(x) * cos(y) with respect to y?

How do you calculate the partial derivative of a composite function such as a^2 * b^3 * c^(1/2) with respect to a?

How do you calculate the partial derivative of a composite function such as a^2 * b^3 * c^(1/2) with respect to b?

How do you calculate the partial derivative of a composite function such as a^2 * b^3 * c^(1/2) with respect to c?