Problem 14 - Calculus Early Transcendentals

Question

Find \(f^{\prime}(x)\) $$ f(x)=\sqrt[3]{\frac{8}{x}} $$

Step-by-Step Solution

Answer

The derivative is \(-\frac{8^{1/3}}{3}x^{-4/3}\).

Step 1: Rewrite the Function

Rewrite the function in a form that is easier to differentiate. Start by expressing the cube root and the fraction in terms of a power function: \[ f(x) = \left(\frac{8}{x}\right)^{\frac{1}{3}} = 8^{\frac{1}{3}}x^{-\frac{1}{3}} \]

Step 2: Apply the Constant Multiple Rule

Differentiate the constant multiple \(8^{\frac{1}{3}}\), leaving it unchanged while applying the power rule to the function. \[ f^{\prime}(x) = 8^{\frac{1}{3}} \cdot \frac{d}{dx} \left(x^{-\frac{1}{3}}\right) \]

Step 3: Apply the Power Rule

Apply the power rule which states that \( \frac{d}{dx}[x^n] = nx^{n-1} \) to get the derivative of \(x^{-\frac{1}{3}}\).\[ \frac{d}{dx}\left(x^{-\frac{1}{3}}\right) = -\frac{1}{3}x^{-\frac{4}{3}} \]

Step 4: Multiply and Simplify

Multiply the result from the power rule by the constant \(8^{\frac{1}{3}}\) to find the derivative:\[ f^{\prime}(x) = 8^{\frac{1}{3}} \cdot \left(-\frac{1}{3}x^{-\frac{4}{3}}\right) = -\frac{8^{\frac{1}{3}}}{3}x^{-\frac{4}{3}} \]This is the simplified version of the derivative.

Key Concepts

Power Rule

The Power Rule is a fundamental concept in differentiation. It provides a simple method to find the derivative of a function involving a term with a variable raised to an exponent. This rule states that if you have a function of the form \( f(x) = x^n \), then the derivative, \( f'(x) \), is \( nx^{n-1} \). This makes calculating derivatives much more manageable when dealing with polynomial functions or functions that can be expressed as powers of \(x\).

In the original exercise, we applied the Power Rule to the term \( x^{-rac{1}{3}} \). By using the Power Rule, we find that the derivative is \( -\frac{1}{3}x^{-rac{4}{3}} \). Here's how it works step-by-step:

• Identify the power \( n \), which is \(-\frac{1}{3}\) in this case.
• Multiply by the power: \( -\frac{1}{3} \times x^{n-1} \).
• Subtract 1 from the power: \( x^{-\frac{1}{3} - 1} = x^{-\frac{4}{3}} \).

So, the Power Rule helps transform the original function into a straightforward derivative.

Constant Multiple Rule

The Constant Multiple Rule is another essential tool in differentiation, especially when you deal with functions that have coefficients. This rule states that if you have a constant multiplied by a function, the derivative of the entire expression is simply the constant multiplied by the derivative of the function. Mathematically, if \( c \) is a constant and \( f(x) \) is a differentiable function, then \( \frac{d}{dx}[c \, f(x)] = c \, f'(x) \).

In the original exercise, we had the constant multiple \( 8^{\frac{1}{3}} \) in our rewritten function. We used the Constant Multiple Rule to handle this constant separately. Here's how you apply it:

• Treat \( 8^{\frac{1}{3}} \) as a constant since it does not depend on \( x \).
• Differentiate the function \( x^{-rac{1}{3}} \) using the Power Rule to find \( f'(x) \).
• Multiply \( 8^{\frac{1}{3}} \) by this derivative: \( 8^{\frac{1}{3}} \times \left(-\frac{1}{3}x^{-rac{4}{3}}\right) \).

This results in the derivative, simplified as \( -\frac{8^{\frac{1}{3}}}{3}x^{-rac{4}{3}} \). By applying the Constant Multiple Rule, we ensured that our constant remained part of the derivative process.

Derivative of a Function

Understanding the Derivative of a Function is crucial for calculus. It represents the rate of change of a function concerning its variable, or the slope of the tangent line to a point on a function's graph. Derivatives are foundational in exploring and understanding changes within various physical and abstract systems.

In the process of solving the original exercise, the derivative \( f'(x) \) was found for a function \( f(x) = \left( \frac{8}{x} \right)^{\frac{1}{3}} \). By first rewriting this function as \( 8^{\frac{1}{3}}x^{-rac{1}{3}} \), and then applying differentiation rules like the Power Rule and Constant Multiple Rule, we deduced the derivative function.

Here's a simplified explanation:

• The original function was not in a convenient form for differentiation. Therefore, it was first rewritten to make it easier to apply the differentiation rules.
• With the differentiation rules, we found the derivative: \( -\frac{8^{\frac{1}{3}}}{3}x^{-rac{4}{3}} \).
• The derivative's result shows how \( f(x) \)'s rate of change varies as \( x \) changes.

Learning to find the derivative of a function through such exercises enhances your comprehension of how functions behave and how calculus helps describe those behaviors.