Let $g(x, Y) = X^2 E^{x \sin(y)}$.Compute The First And Second Partial Derivatives And Label Them With The Notation You Like Best.

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Let's Dive into Partial Derivatives: Computing g(x, y) and its Derivatives

In this article, we will explore the concept of partial derivatives and apply it to the function g(x, y) = x^2 e^(x sin(y)). We will compute the first and second partial derivatives of this function and label them using the notation we prefer. This will help us understand the behavior of the function and its response to changes in the variables x and y.

The function g(x, y) is a composite function that involves the exponential function and the sine function. It is defined as:

g(x, y) = x^2 e^(x sin(y))

To compute the first partial derivatives of g(x, y), we will differentiate the function with respect to x and y separately.

Partial Derivative with Respect to x

To find the partial derivative of g(x, y) with respect to x, we will treat y as a constant and differentiate the function with respect to x.

∂g/∂x = ∂(x^2 e^(x sin(y)))/∂x

Using the product rule and the chain rule, we get:

∂g/∂x = 2xe^(x sin(y)) + x^2 e^(x sin(y)) sin(y)

Simplifying the expression, we get:

∂g/∂x = x^2 e^(x sin(y)) (2 + sin(y))

Partial Derivative with Respect to y

To find the partial derivative of g(x, y) with respect to y, we will treat x as a constant and differentiate the function with respect to y.

∂g/∂y = ∂(x^2 e^(x sin(y)))/∂y

Using the chain rule, we get:

∂g/∂y = x^2 e^(x sin(y)) x cos(y)

Simplifying the expression, we get:

∂g/∂y = x^3 e^(x sin(y)) cos(y)

To compute the second partial derivatives of g(x, y), we will differentiate the first partial derivatives with respect to x and y separately.

Second Partial Derivative with Respect to x

To find the second partial derivative of g(x, y) with respect to x, we will differentiate the first partial derivative with respect to x.

∂²g/∂x² = ∂(x^2 e^(x sin(y)) (2 + sin(y)))/∂x

Using the product rule and the chain rule, we get:

∂²g/∂x² = 2x e^(x sin(y)) (2 + sin(y)) + x^2 e^(x sin(y)) (2 + sin(y)) sin(y) + x^2 e^(x sin(y)) (2 + sin(y)) sin(y)

Simplifying the expression, we get:

∂²g/∂x² = x^2 e^(x sin(y)) (2 + sin(y)) (2 + 2 sin(y))

Second Partial Derivative with Respect to y

To find the second partial derivative of g(x, y) with respect to y, we will differentiate the first partial derivative with respect to y.

∂²g/∂y² = ∂(x^3 e^(x sin(y)) cos(y))/∂y

Using the chain rule, we get:

∂²g/∂y² = x^3 e^(x sin(y)) (-x sin(y) cos(y)) + x^3 e^(x sin(y)) cos²(y)

Simplifying the expression, we get:

∂²g/∂y² = x^3 e^(x sin(y)) (-x sin(y) cos(y) + cos²(y))

Second Partial Derivative with Respect to x and y

To find the second partial derivative of g(x, y) with respect to x and y, we will differentiate the first partial derivative with respect to x and y.

∂²g/∂x∂y = ∂(x^2 e^(x sin(y)) (2 + sin(y)))/∂y

Using the chain rule, we get:

∂²g/∂x∂y = x^2 e^(x sin(y)) x cos(y) (2 + sin(y))

Simplifying the expression, we get:

∂²g/∂x∂y = x^3 e^(x sin(y)) cos(y) (2 + sin(y))

In this article, we have computed the first and second partial derivatives of the function g(x, y) = x^2 e^(x sin(y)). We have labeled the derivatives using the notation we prefer and have simplified the expressions to make them easier to understand. This will help us understand the behavior of the function and its response to changes in the variables x and y.

  • [1] "Partial Derivatives" by Wolfram MathWorld
  • [2] "Partial Derivatives" by Khan Academy
  • [3] "Partial Derivatives" by MIT OpenCourseWare
  • [1] "Multivariable Calculus" by James Stewart
  • [2] "Calculus" by Michael Spivak
  • [3] "Differential Equations" by Lawrence Perko

In our previous article, we explored the concept of partial derivatives and applied it to the function g(x, y) = x^2 e^(x sin(y)). We computed the first and second partial derivatives of this function and labeled them using the notation we prefer. In this article, we will answer some frequently asked questions about partial derivatives and the function g(x, y).

A: A partial derivative is the derivative of a function with respect to one of its variables, while treating the other variables as constants. A total derivative, on the other hand, is the derivative of a function with respect to all of its variables.

A: To compute the partial derivative of a function with respect to x and y, you will need to differentiate the function with respect to x and y separately. This will give you two partial derivatives, ∂f/∂x and ∂f/∂y.

A: The notation for partial derivatives is ∂f/∂x, where f is the function and x is the variable with respect to which the derivative is being taken.

A: To compute the second partial derivative of a function with respect to x and y, you will need to differentiate the first partial derivative with respect to x and y. This will give you four second partial derivatives, ∂²f/∂x², ∂²f/∂y², ∂²f/∂x∂y, and ∂²f/∂y∂x.

A: The first partial derivatives of a function are used to compute the second partial derivatives. The second partial derivatives can be used to determine the concavity of the function and the location of its inflection points.

A: Partial derivatives can be used to optimize a function by finding the critical points of the function. The critical points are the points where the first partial derivatives are equal to zero. The second partial derivatives can be used to determine the nature of the critical points.

A: Partial derivatives have many applications in physics, engineering, economics, and other fields. Some common applications include:

  • Finding the maximum and minimum values of a function
  • Determining the concavity of a function
  • Finding the inflection points of a function
  • Optimizing functions
  • Modeling real-world phenomena

In this article, we have answered some frequently asked questions about partial derivatives and the function g(x, y). We hope that this article has been helpful in clarifying the concepts of partial derivatives and their applications.

  • [1] "Partial Derivatives" by Wolfram MathWorld
  • [2] "Partial Derivatives" by Khan Academy
  • [3] "Partial Derivatives" by MIT OpenCourseWare
  • [1] "Multivariable Calculus" by James Stewart
  • [2] "Calculus" by Michael Spivak
  • [3] "Differential Equations" by Lawrence Perko

Note: The references and further reading section are not exhaustive and are provided for additional information and resources.