Problem 1 - Calculus Single Variable

Question

In each of Exercises \(1-6, y\) is a function of \(x .\) Calculate the derivative of the given expression with respect to \(x\). (Your answer should contain the term \(d y / d x .)\) \(4 y^{3 / 2}\)

Step-by-Step Solution

Answer

The derivative is \(6y^{1/2} \frac{dy}{dx}\).

Step 1: Understand the Problem

We need to find the derivative of the expression \(4y^{3/2}\) with respect to \(x\). Since \(y\) is a function of \(x\), we'll apply implicit differentiation.

Step 2: Differentiate with Respect to x

Differentiate the entire expression \(4y^{3/2}\) with respect to \(x\). Use the chain rule since we have \(y\) as a function of \(x\).

Step 3: Apply the Chain Rule

Use the chain rule: if \(u = y^{3/2}\), then \(\frac{d}{dx}[4u] = 4 \cdot \frac{d}{dx}[u] = 4 \cdot \frac{3}{2} y^{1/2} \cdot \frac{dy}{dx}\). This gives us \(6y^{1/2} \cdot \frac{dy}{dx}\).

Step 4: Present the Derivative

The derivative of the expression \(4y^{3/2}\) with respect to \(x\) is \(6y^{1/2} \frac{dy}{dx}\).

Key Concepts

Chain Rule in Implicit Differentiation

The chain rule is a fundamental concept used when differentiating composite functions. It becomes a vital tool in implicit differentiation, especially when dealing with expressions where one variable is dependent on another, like in our exercise. When you encounter a function like \( 4y^{3/2} \), and you know that \( y \) is a function of \( x \), you cannot differentiate it normally. Here, you apply the chain rule.

So, imagine you have a function \( u = y^{3/2} \). To find the derivative of this in terms of \( x \), first, differentiate \( u \) with respect to \( y \), then multiply the result by \( \frac{dy}{dx} \), which is the derivative of \( y \) with respect to \( x \). Essentially, the chain rule tells you:

• If \( u = y^{3/2} \), then \( \frac{d}{dy}[u] = \frac{3}{2} y^{1/2} \).
• Then, the derivative \( \frac{d}{dx}[4u] = 4 \cdot \frac{3}{2} y^{1/2} \cdot \frac{dy}{dx} \).

This step-by-step application is what gives the chain rule its power in solving implicit differentiation problems.

How to Calculate the Derivative

Derivative calculation involves finding out how a function changes as its input changes. In this context, with functions of \( x \) and \( y \), it becomes a bit more involved. Calculating the derivative of \( 4y^{3/2} \) requires understanding that \( y \) itself changes with \( x \).

Start by differentiating the constant multiplier with the chain rule:

• Differentiate \( 4y^{3/2} \) with respect to \( y \) to isolate the inner derivative \( 4 \cdot \frac{3}{2} y^{1/2} \).
• The factor of \( \frac{dy}{dx} \) accounts for the fact that \( y \) is actually changing.

This combination of moving parts is how you derive \( 6y^{1/2} \frac{dy}{dx} \). This simplification helps you understand how each part of the expression contributes to its rate of change.

Exploring Functions of x and y

In many math problems, especially those involving calculus, functions can be intertwined, with one depending on the other. A function of \( y \) like \( y^{3/2} \), where \( y \) itself is a function of \( x \), is a prime example of such cases. These are often tackled through implicit differentiation.

Implicit differentiation is employed here because \( y \) is not simple; it varies as \( x \) changes. Through this method, you acknowledge and handle the dependency without needing to express \( y \) solely in terms of \( x \). This approach helps calculate derivatives without rearranging equations extensively.

Consider the expression again: \( 4y^{3/2} \). We first acknowledge that \( y \) changes with \( x \) and differentiate accordingly. This approach elegantly solves problems where direct differentiation isn't straightforward due to interdependent variables, maintaining the integrity of the relationships between \( x \) and \( y \).