Problem 3 - Thomas Calculus

Question

Find \(d y / d x\) in Exercises \(1-10\) $$ y=\sqrt[3]{2 x} $$

Step-by-Step Solution

Answer

The derivative is \( \frac{1}{3 \cdot 2^{1/3} x^{2/3}} \).

Step 1: Understand the Function

The given function is \( y = \sqrt[3]{2x} \). This is equivalent to writing \( y = (2x)^{1/3} \). Recognizing this helps us apply proper differentiation rules.

Step 2: Apply the Chain Rule

To find \( \frac{dy}{dx} \), we need to differentiate \( y = (2x)^{1/3} \). We'll use the chain rule here. The chain rule states that if \( y = (u)^{n} \), then \( \frac{dy}{dx} = n \cdot (u)^{n-1} \cdot \frac{du}{dx} \). For our function, \( u = 2x \) and \( n = \frac{1}{3} \).

Step 3: Differentiate the Outer Function

Differentiate the outer function \( (u)^{1/3} \) with respect to \( u \). This gives us \( \frac{d}{du}(u)^{1/3} = \frac{1}{3} u^{-2/3} \).

Step 4: Differentiate the Inner Function

Differentiate the inner function \( u = 2x \) with respect to \( x \). This gives us \( \frac{du}{dx} = 2 \).

Step 5: Combine Derivatives

Using the results from Steps 3 and 4, combine them to get \( \frac{dy}{dx} = \frac{1}{3} (2x)^{-2/3} \cdot 2 = \frac{2}{3} (2x)^{-2/3} \).

Step 6: Simplify the Expression

The derivative \( \frac{dy}{dx} = \frac{2}{3} (2x)^{-2/3} \) can be simplified to \( \frac{2}{3} \cdot \frac{1}{(2x)^{2/3}} \) which becomes \( \frac{2}{3} \cdot \frac{1}{2^{2/3} x^{2/3}} \). Finally, simplify the constant term to get the final derivative: \( \frac{1}{3 \cdot 2^{1/3} x^{2/3}} \).

Key Concepts

Chain Rule

The Chain Rule is an essential tool in Calculus when you want to differentiate compositions of functions. Imagine you have a situation where two functions are combined, and you need to understand how one changes in response to changes in another. This is precisely where the Chain Rule becomes invaluable.
In basic terms, if you have a function composition such as \( y = f(g(x)) \), the chain rule allows you to differentiate this by multiplying the derivative of the outer function \( f \) with the derivative of the inner function \( g \). Here's the formula:

• Identify the inner function: \( u = g(x) \)
• Identify the outer function: \( y = f(u) \)
• Then: \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \)

This step-by-step process ensures you do not miss any part of the differentiation of such nested functions.

For our example, the inner function is \( u = 2x \), and the outer function is \( y = u^{1/3} \). After finding their derivatives, we multiply them to find the derivative of the whole function.

Power Rule

The Power Rule is your go-to shortcut for finding derivatives of expressions where a variable is raised to a constant power. This rule states that if \( y = x^n \), the derivative \( \frac{dy}{dx} = n \cdot x^{n-1} \). It is a straightforward approach and speeds up the differentiation process significantly.

In the practice exercise, understanding the power rule is critical because the expression \( (2x)^{1/3} \) can be expanded as \( u^{1/3} \), where the variables are raised to fractional powers.

Here's how you apply it:

• Identify the power \( n \): In \( (2x)^{1/3} \), \( n = \frac{1}{3} \).
• Differentiate using the power rule formula \( n \cdot x^{n-1} \).

Applying these steps results in finding the derivative of the outer function. The Power Rule saves time and helps simplify complex algebraic operations involving powers of variables.

Fractional Exponents

Fractional exponents may seem daunting, but they are merely another way to express roots or powers of variables. When you encounter an expression like \( \sqrt[3]{2x} \), it can be written as \( (2x)^{1/3} \).

Understanding fractional exponents is essential for applying rules of differentiation, such as the power rule and chain rule. Let's break it down:

• The numerator of the fraction indicates the power, which tells us how many times to multiply the base by itself.
• The denominator is the root that tells us which root of the number we're taking.

Fractional exponents provide a uniform way to handle roots and powers, particularly useful when differentiating functions with roots.

In our exercise, recognizing that \( \sqrt[3]{2x} \) is the same as \( (2x)^{1/3} \) allows us to apply the Power Rule directly. This simplifies otherwise complex calculations and makes solving calculus problems more manageable.