Problem 6 - Calculus Early Transcendentals

Question

Suppose \(F(x, y)=0\) and \(y\) is a differentiable function of \(x\). Explain how to find \(d y / d x\).

Step-by-Step Solution

Answer

To find the derivative of y with respect to x, or dy/dx, we can use the concept of implicit differentiation. The derivative is found by taking the negative ratio of the partial derivatives of the function F(x, y) with respect to x and y, as follows: dy/dx = - (∂F/∂x) / (∂F/∂y) This method allows us to calculate the derivative without explicitly solving for y as a function of x.

Step 1: Differentiate both sides of the function with respect to x

We start by taking the derivative of both sides of the function \(F(x, y) = 0\) with respect to \(x\). Note that we must apply the chain rule to differentiate with respect to y, since y is a function of x. We will write this step as \(\frac{d}{dx}(F(x, y))=\frac{d}{dx}(0)\).

Step 2: Apply the chain rule to differentiate with respect to y

Now we apply the chain rule to differentiate the function \(F(x, y)\) with respect to y. Rewriting \(\frac{d}{dx}(F(x, y))\) as \(\frac{\partial F}{\partial x} + \frac{\partial F}{\partial y} \frac{dy}{dx}\), since \(y\) is a function of \(x\).

Step 3: Substitute and solve for the derivative

Now, we can substitute the derivative on both sides of the equation. We have \(\frac{\partial F}{\partial x} + \frac{\partial F}{\partial y} \frac{dy}{dx} = 0\). Solving for \(\frac{dy}{dx}\), we can rewrite the equation as \(\frac{dy}{dx} = -\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial y}}\).

Step 4: Interpret the result

Thus, the derivative \(\frac{dy}{dx}\) can be found by taking the negative ratio of the partial derivatives of the function \(F(x, y)\) with respect to \(x\) and \(y\). This approach using implicit differentiation allows us to find the derivative of y with respect to x without explicitly solving for y as a function of x.

Key Concepts

Partial Derivatives

In calculus, partial derivatives are an essential tool, especially when dealing with functions that have more than one variable. When you have a function like \( F(x, y) \), which involves the variables \( x \) and \( y \), partial derivatives allow you to focus on the change of the function relative to one variable while keeping the others constant.

For example, \( \frac{\partial F}{\partial x} \) represents the rate of change of the function \( F \) with respect to \( x \), treating \( y \) as if it were constant. Similarly, \( \frac{\partial F}{\partial y} \) shows how \( F \) changes concerning \( y \), assuming \( x \) was fixed.

Partial derivatives are crucial in implicit differentiation. Because \( y \) is expressed implicitly in terms of \( x \), we rely on these derivatives to uncover how changes in \( x \) are associated with changes in \( y \). Utilizing partial derivatives simplifies handling multi-variable functions and explores the function's dynamics thoroughly.

Chain Rule

The chain rule is a fundamental concept used to compute the derivative of composite functions. It’s particularly useful when differentiating functions where one variable is dependent on another, as in implicit differentiation.

To understand how it works, imagine you have a nested function where \( y \) is a function of \( x \), and it influences another function \( F(x, y) \). Here, the chain rule helps differentiate \( F \) with respect to \( x \) by also considering how \( y \) changes as \( x \) changes.

In our exercise, when tackling \( \frac{d}{dx}(F(x, y)) \), the chain rule allows us to rewrite the expression as \( \frac{\partial F}{\partial x} + \frac{\partial F}{\partial y} \frac{dy}{dx} \). This breaks down the differentiation process into simpler pieces, making it easier to find \( \frac{dy}{dx} \) using information from both \( \frac{\partial F}{\partial x} \) and \( \frac{\partial F}{\partial y} \).

By effectively applying the chain rule, we gain a structured way to handle derivatives of more complex functions, shedding light on the interaction between variables.

Differentiable Function

A differentiable function is one that allows us to compute a derivative at every point within its domain. This property is vital for solving many problems in calculus, as it ensures the function behaves predictably and smoothly.

In the context of implicit differentiation, knowing that \( y \) is a differentiable function of \( x \) assures us that we can determine \( \frac{dy}{dx} \). It means that despite not having an explicit expression for \( y \) in terms of \( x \), \( y\) changes at a consistent rate with \( x \).

Differentiability is what makes the implicit differentiation technique applicable—it’s the foundation that supports using partial derivatives and the chain rule seamlessly. Without differentiability, we wouldn't be able to trust the derivative calculations or the relationships they reveal. Thus, understanding a function's differentiable nature confirms its readiness for exploration with tools like the chain rule and implicit differentiation.