Finding the Gradient of a Multivariate Implicit Function

In the field of mathematics, particularly in calculus, implicit differentiation is a powerful tool for finding the derivatives of functions that are not expressed in the form y f(x). Let's explore how to find the gradient of the multivariate implicit function ( x^2 y^2 - 1 x^2y^3 ) through step-by-step examples and detailed explanations.

Introduction to Implicit Differentiation

Implicit differentiation is the process of finding the derivative of a function that is defined implicitly by an equation involving both the dependent and independent variables. This technique is particularly useful when the relationship between the variables is not expressed in a simple explicit form. In this article, we will apply implicit differentiation to find the gradient of an implicit function, which is essentially the partial derivatives with respect to both (x) and (y).

The Function and the Implicit Equation

Consider the following multivariate implicit function:

[ x^2 y^2 - 1 x^2y^3 ]

Our goal is to find the gradient of this function, which means we need to find the partial derivatives with respect to (x) and (y).

Differentiating Implicitly

To find the gradient, we start by differentiating both sides of the equation with respect to (x), keeping in mind that (y) is a function of (x). This process involves treating (y) as a dependent variable and applying the chain rule.

Step 1: Differentiate with Respect to (x)

First, let's differentiate the left side of the equation with respect to (x):

[ frac{d}{dx}(x^2 y^2 - 1) 2x 2yfrac{dy}{dx} - 0 ]

Now, let's differentiate the right side of the equation with respect to (x):

[ frac{d}{dx}(x^2y^3) 2xy^3 x^2 cdot 3y^2 frac{dy}{dx} ]

Putting these together, we get:

[ 2x 2yfrac{dy}{dx} 2xy^3 3x^2y^2 frac{dy}{dx} ]

Next, we isolate the term involving (frac{dy}{dx}) (the gradient with respect to (x)):

[ 2yfrac{dy}{dx} - 3x^2y^2 frac{dy}{dx} 2xy^3 - 2x ]

Factoring out (frac{dy}{dx}) from the left side:

[ (2y - 3x^2y^2) frac{dy}{dx} 2x(1 - y^2) ]

Solving for (frac{dy}{dx}):

[ frac{dy}{dx} frac{2x(1 - y^2)}{2y - 3x^2y^2} ]

We can simplify this further if possible, but for now, we have the gradient with respect to (x).

Step 2: Substitute the Given Values

We are given that (x 1) and (y 0). Substituting these values into the expression for (frac{dy}{dx}):

[ frac{dy}{dx} frac{2(1)(1 - 0^2)}{2(0) - 3(1)^2(0)^2} frac{2}{0} ]

This leads to a division by zero, which indicates that the gradient is undefined at ((1, 0)).

Conclusion and Extensions

In this article, we walked through the process of finding the gradient of a multivariate implicit function ( x^2 y^2 - 1 x^2y^3 ) using implicit differentiation. We saw how to differentiate the equation with respect to both (x) and (y), and then evaluated the gradient at a specific point ((1, 0)), where the gradient was undefined.

Implicit differentiation is a versatile method that extends beyond this simple example. Understanding its application in various contexts can help in solving more complex multivariate functions and systems of equations. It is a valuable tool in fields such as engineering, physics, and economics, where relationships between multiple variables are common.

For further exploration, consider studying other multivariate functions and practicing implicit differentiation with different equations. This will help solidify your understanding and provide new insights into mathematical relationships.