Problem 7 - Calculus for Biology and Medicine

Question

In Problems \(1-8\), find \(\frac{d y}{d x}\) by implicit differentiation. $$ \frac{x}{y}=\frac{y}{x} $$

Step-by-Step Solution

Answer

The derivative \( \frac{dy}{dx} = \frac{x}{y} \).

Step 1: Express Given Equation

The given equation is \( \frac{x}{y} = \frac{y}{x} \). This equation will be differentiated implicitly with respect to \( x \).

Step 2: Cross Multiply

To eliminate the fractions, cross multiply the equation: \( x^2 = y^2 \).

Step 3: Differentiate Both Sides

Differentiate both sides with respect to \( x \). The derivative of \( x^2 \) is \( 2x \), and the derivative of \( y^2 \) using the chain rule is \( 2y \frac{dy}{dx} \), since \( y \) is a function of \( x \).

Step 4: Collect Derivative Terms

Set up the equation from the derivatives: \( 2x = 2y \frac{dy}{dx} \).

Step 5: Solve for \( \frac{dy}{dx} \)

Divide both sides by \( 2y \) to solve for \( \frac{dy}{dx} \): \( \frac{dy}{dx} = \frac{x}{y} \).

Key Concepts

calculus

Calculus is a branch of mathematics that studies how things change. At its core, calculus deals with the concepts of rates of change (derivatives) and accumulation (integrals). In this exercise, we're focused on derivatives, which essentially measure how a function changes as its input changes.
Understanding calculus involves comprehending how different functions behave and how to approach problems involving those functions. In implicit differentiation, like our problem involves, we're finding the rate of change of one variable with respect to another when they're not easily separated into one variable per side of the equation.

• Implicit Differentiation: Used when you have equations involving two variables intertwined in a more complex way.
• Normal Differentiation: Preferred when you can easily express one variable solely as a function of the other.

Calculus allows us to analyze scenarios where direct calculation is difficult by abstracting the problem and finding solutions through differentiation rules.

chain rule

The Chain Rule is a fundamental tool in calculus that helps us differentiate composite functions. It's especially useful in situations involving implicit differentiation. When you have a function nested within another function, the chain rule is your go-to method for finding the derivative.
Here's how it works in the context of our exercise: When differentiating the term \(y^2\), you treat it as a composite function because \(y\) is a function of \(x\). Instead of just taking the derivative of \(y^2\) directly, you perform the following steps:

• Differentiate the outer function, which in this case is \(z^2\), giving \(2z\).
• Multiply the result by the derivative of the inner function, \(rac{dy}{dx}\).

This gives you \(2y \rac{dy}{dx}\), which is a perfect demonstration of how the chain rule operates to help solve derivatives when variables depend on each other.
This rule is integral in implicit differentiation where a direct approach isn't possible.

derivatives

Derivatives are the heart of calculus describing how a function changes as its input changes. In practical terms, the derivative of a function at a point tells you the slope of the tangent line to the curve at that point.
Different forms of derivatives can exist depending on how the function variables are presented:

• Explicit Differentiation: The straightforward method applied when one variable is isolated.
• Implicit Differentiation: Unused when variables are not isolated, and need to be differentiated as they appear.

In the problem, the derivative of \(x^2\) was straightforward, but \(y^2\) required \(y\) be treated as a differentiable function of \(x\), leading to the form \(2y \frac{dy}{dx}\). This illustrates how derivative formulas and rules transform complex function equations into solvable pieces.

cross multiplication

Cross multiplication is a mathematical technique used to simplify an equation with fractions and eliminate related complexities. In our problem, solving \(\frac{x}{y} = \frac{y}{x}\) directly would be challenging without transforming it first.
By cross-multiplying, you multiply both sides of the equation to remove the fractions, giving you a simplified equation without denominators. Here’s how it works for our problem:

• Initially, you have \(\frac{x}{y} = \frac{y}{x}\).
• By multiplying across the equals sign, you get \(x \cdot x = y \cdot y\).
• This simplifies to \(x^2 = y^2\).

Cross multiplication transforms a complex fraction-based equation into a simpler one that can be directly differentiated.
It is a crucial step for handling equations where variables interact in a multiplicative context.