Problem 36 - Technical Mathematics with Calculus

Question

Write the differential \(d y\) in terms of \(x, y,\) and \(d x\) for each implicit relation. $$x^{3}+2 y^{3}=5$$

Step-by-Step Solution

Answer

The differential dy in terms of x, y, and dx is \frac{dy}{dx} = \frac{-3x^{2}}{6y^{2}} dx, hence dy = \frac{-3x^{2}}{6y^{2}} dx.

Step 1: Differentiate Implicitly

Differentiate both sides of the equation with respect to x. Remember to apply the chain rule to differentiate y with respect to x.

Step 2: Apply the Chain Rule

When differentiating the term involving y, since y is a function of x, you will need to multiply by the derivative of y with respect to x, denoted as dy/dx.

Step 3: Solve for dy/dx

Rearrange the equation to solve for the derivative dy/dx.

Step 4: Express dy

Since dy/dx is the rate of change of y with respect to x, express dy in terms of dx by multiplying both sides of the equation by dx.

Key Concepts

Differential Calculus

Differential calculus is a branch of mathematics that deals with the study of rates at which quantities change. It's the foundation for understanding the concept of a derivative, which measures how a function's output value changes in response to changes in its input value. In a mathematical function, you have independent variables (like the variable 'x') and dependent variables (whose value depends on the independent variable, like 'y'). The rate of change of these variables with respect to each other is crucial in many fields, including physics, engineering, and economics.

For example, if you speed up or slow down your car, differential calculus helps to measure how quickly the velocity is changing over time. Similarly, in our exercise involving the implicit relation \(x^{3}+2 y^{3}=5\), differential calculus enables us to find the rate at which \(y\) changes with respect to \(x\), denoted as \(dy/dx\).

Chain Rule

The chain rule is a fundamental theorem in calculus used to take the derivative of composite functions. When functions are combined in such a way that the output of one becomes the input of another, the chain rule enables us to differentiate the entire combination with respect to the main independent variable.

The classic form of the chain rule is expressed as \(\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)\), where \(f\) and \(g\) are functions of \(x\). In the context of our exercise, when we encounter a term like \(2y^{3}\), we apply the chain rule to differentiate it with respect to \(x\). Since \(y\) is implicitly a function of \(x\), the derivative of \(2y^{3}\) with respect to \(x\) is \(6y^{2}\frac{dy}{dx}\), signifying the derivative of the outside function \(2y^{3}\) times the derivative of the inside function \(y\).

Derivative of Implicit Functions

Implicit differentiation is a technique used to find the derivative of functions that are not stated directly as one variable in terms of another, hence 'implicit' in contrast to 'explicit' functions. It allows us to differentiate equations involving two or more variables without having to explicitly solve for one variable in terms of the others.

In our exercise, we have the equation \(x^{3}+2y^{3}=5\). It is not explicitly solved for \(y\), and yet we are tasked with finding the differential \(dy\) with respect to \(x\). We achieve this by differentiating each term with respect to \(x\) and applying the chain rule, as already mentioned, when differentiating terms involving \(y\). The end goal is to isolate \(dy/dx\) and then express \(dy\) as a function of \(x\), \(y\), and \(dx\), which represents a small change in \(x\). This process encapsulates the essence of differential calculus and highlights the power of the chain rule in handling derivatives of implicit functions.