Problem 3 - Calculus for Biology and Medicine

Question

In Problems \(1-8\), find \(\frac{d y}{d x}\) by implicit differentiation. $$ x^{3 / 4}+y^{3 / 4}=1 $$

Step-by-Step Solution

Answer

\( \frac{d y}{d x} = -\frac{y^{1/4}}{x^{1/4}} \)

Step 1: Differentiate both sides with respect to x

Start by differentiating both sides of the equation \( x^{3/4} + y^{3/4} = 1 \) with respect to \( x \). Since the right-hand side is constant, its derivative is zero. For the left-hand side, we use implicit differentiation.

Step 2: Apply Chain Rule to y-term

When differentiating \( y^{3/4} \) with respect to \( x \), use the chain rule: the derivative of \( y^{3/4} \) is \( \frac{3}{4} y^{-1/4} \cdot \frac{d y}{d x} \).

Step 3: Differentiate x-term

The derivative of \( x^{3/4} \) with respect to \( x \) is calculated as \( \frac{3}{4} x^{-1/4} \).

Step 4: Set up equation

From the previous steps, set up the equation: \( \frac{3}{4} x^{-1/4} + \frac{3}{4} y^{-1/4} \cdot \frac{d y}{d x} = 0 \).

Step 5: Solve for \( \frac{d y}{d x} \)

Re-arrange the equation to solve for \( \frac{d y}{d x} \). First, subtract \( \frac{3}{4} x^{-1/4} \) from both sides yielding \( \frac{3}{4} y^{-1/4} \cdot \frac{d y}{d x} = -\frac{3}{4} x^{-1/4} \). Then, divide both sides by \( \frac{3}{4} y^{-1/4} \), leading to \( \frac{d y}{d x} = -\frac{3 x^{-1/4}}{3 y^{-1/4}} = -\frac{y^{1/4}}{x^{1/4}} \).

Key Concepts

Partial Derivatives

When dealing with functions of multiple variables, partial derivatives are a fundamental tool. They allow us to understand how a function changes as one variable changes while others are constant. In implicit differentiation, partial derivatives help differentiate terms with respect to a specific variable, like separating effects of one variable from another.

Consider the equation we are given:

• \( x^{3/4} + y^{3/4} = 1 \)

When differentiating, use the concept of partial derivatives to handle each term separately:
- For the \( x \) term, derive it directly with respect to \( x \).
- For the \( y \) term, treat \( y \) as a dependent variable and apply the chain rule.

This approach helps us calculate the derivative with precision, accounting for all variables appropriately.

Chain Rule

The chain rule is essential when differentiating composite functions. It gives a way to find derivatives for expressions where one variable depends on another. In the process of implicit differentiation, the chain rule helps with terms like \( y^{3/4} \) because \( y \) is a function of \( x \).

To apply the chain rule to \( y^{3/4} \), follow these steps:

• First, find the derivative of the outside function, \( y^{3/4} \), with respect to \( y \). This is \( \frac{3}{4}y^{-1/4} \).
• Then, multiply by the derivative of \( y \) with respect to \( x \), which is \( \frac{d y}{d x} \).

This yields the term \( \frac{3}{4}y^{-1/4} \cdot \frac{d y}{d x} \), showing the application of the chain rule in handling functions of another variable.

Solving Equations

After finding derivatives, the next step involves manipulating the equation to solve for the desired derivative, \( \frac{d y}{d x} \).

Here's a walkthrough of how to solve the given equation:

• Start with the differentiated equation: \( \frac{3}{4} x^{-1/4} + \frac{3}{4} y^{-1/4} \cdot \frac{d y}{d x} = 0 \).
• Isolate the \( \frac{d y}{d x} \) term by subtracting the \( x \) component from both sides: \( \frac{3}{4} y^{-1/4} \cdot \frac{d y}{d x} = -\frac{3}{4} x^{-1/4} \).
• Divide through by \( \frac{3}{4} y^{-1/4} \) to solve for \( \frac{d y}{d x} \), which gives: \( \frac{d y}{d x} = -\frac{y^{1/4}}{x^{1/4}} \).

This process involves simple algebraic manipulation, crucial for obtaining the final expression for the derivative.

Calculus

Calculus is the mathematical study of change, where derivatives represent how a function varies as its inputs vary. Implicit differentiation, a calculus technique, is used when functions are not easily separated into explicit forms. It is especially valuable in solving equations like the one given, where \( y \) cannot be easily isolated.

By using implicit differentiation, we can:

• Differentiate both sides of the equation with respect to \( x \).
• Apply necessary derivative rules, such as the chain rule, to handle composite terms.
• Rearrange the resulting terms to find the implicit relation that describes how \( y \) changes with respect to \( x \).

Overall, calculus provides powerful tools for exploring and understanding relationships within mathematical models and real-world phenomena.