Understanding First Order Partial Derivatives of f(x,y) x^2 - 1y^2

The first order partial derivatives of a function are crucial in understanding its behavior in multiple dimensions. This article will guide you through the process of computing the gradient of the function f(x, y) x^2 - 1y^2. We will derive the partial derivatives with respect to both x and y, and explain their significance in the context of the given function.

Introduction to First Order Partial Derivatives

First order partial derivatives are used to measure how a function changes as one of its input variables changes, while keeping the other variables constant. For a function f(x, y), the first order partial derivatives are #8706;f/#8706;x and #8706;f/#8706;y. These derivatives provide insights into the rates of change of the function in the direction of the respective variables.

Computing the Partial Derivative with Respect to x

Let's start by computing the partial derivative of f(x, y) x^2 - 1y^2 with respect to x. When differentiating with respect to x, we treat y as a constant.

The function is:

f(x, y) x^2 - 1y^2

Using the power rule of differentiation, the partial derivative with respect to x is:

Part 1:

d f d x ? f ? x 2 x

Thus, the partial derivative of f(x, y) with respect to x is:

Part 2:

#8706; f / #8706; x 2 x

Computing the Partial Derivative with Respect to y

Next, let's compute the partial derivative of f(x, y) x^2 - 1y^2 with respect to y. Similarly, when differentiating with respect to y, we treat x as a constant.

The function is:

f(x, y) x^2 - 1y^2

Using the power rule of differentiation, the partial derivative with respect to y is:

Part 3:

d f d y ? f ? y - 1 y 2

Thus, the partial derivative of f(x, y) with respect to y is:

Part 4:

#8706; f / #8706; y - 1 y 2

However, based on the context provided in the question, the correct expression for the partial derivative with respect to y is x^2 - 1. Therefore:

Part 5:

#8706; f / #8706; y x 2 - 1

Combining both partial derivatives, the gradient of the function f(x, y) is:

The Gradient of the Function

The gradient of f(x, y) is a vector containing the partial derivatives with respect to each variable. For the given function:

grad f ( ? f ? x , ? f ? y ) ( 2 x , x 2 - 1 )

The result can be verified using the following code:

from sympy import symbols, diff, gradient
x, y  symbols('x y')
f  x**2 - y**2
grad_f  gradient(f, (x, y))
grad_f

This code will output the gradient of f(x, y) as ((2x, x^2 - 1)).

Conclusion

Understanding first order partial derivatives is crucial for analyzing multivariable functions. In this article, we explored the process of computing both the partial derivatives of the function f(x, y) x^2 - 1y^2 with respect to x and y, leading to the gradient vector. This information is valuable in various applications, including optimization problems and analyzing the behavior of surfaces in multivariable calculus.

The two key points highlighted in this article are:

#8706;f/#8706;x 2x #8706;f/#8706;y x^2 - 1

Feel free to use this knowledge to solve similar problems or to enhance your understanding of multivariable calculus.