Problem 34 - Brief Applied Calculus

Question

31-38. Find the indicated derivatives. If \(f(x)=12 \sqrt[3]{x^{2}}+\frac{48}{\sqrt[3]{x}}\), find \(f^{\prime}(8)\).

Step-by-Step Solution

Answer

\( f'(8) = 3 \).

Step 1: Find f'(x)

\( f(x) = 12x^{2/3} + 48x^{-1/3} \).
\( f'(x) = 12 \cdot \frac{2}{3}x^{-1/3} + 48 \cdot (-\frac{1}{3})x^{-4/3} = 8x^{-1/3} - 16x^{-4/3} \).

Step 2: Evaluate at x=8

\( 8^{1/3} = 2 \), so \( 8^{-1/3} = 1/2 \) and \( 8^{-4/3} = 1/16 \).
\( f'(8) = 8(1/2) - 16(1/16) = 4 - 1 = 3 \).

Key Concepts

Differentiation Techniques

Differentiation is a fundamental operation in calculus that allows us to find the rate at which a function is changing at any point. The process of finding a derivative involves applying various techniques that simplify the work based on the form of the function. Let’s go over a few key techniques that come up frequently:

• Direct Differentiation: Use basic rules such as the power rule, product rule, quotient rule, and chain rule depending on the function.
• Function Simplification: Simplify the function algebraically first, if needed. This reduces errors and makes applying differentiation rules easier.
• Multiple Terms: Differentiate terms separately from a sum or difference. Each term can often be handled with specific rules.

Before tackling a differentiation problem, analyze the function to determine the appropriate technique. In our exercise, because the function involves exponents and roots, direct simplification and the power rule are both effective strategies.

Power Rule

The power rule is one of the most straightforward and widely used rules in differentiation. It allows us to differentiate functions of the form \(x^n\), where \(n\) is any real number. This rule states that if \(f(x) = x^n\), then the derivative \(f'(x) = nx^{n-1}\).

To apply this rule, simply follow these steps:

• Identify the power \(n\) in your term \(x^n\).
• Multiply by the power: \(n \cdot x^{n-1}\).
• Decrease the exponent by one.

In our original exercise, we transformed the expression into a form where the power rule could be applied easily. For example, \(x^{2/3}\) and \(x^{-1/3}\) were prime candidates for the power rule. We would calculate the derivatives using these simplified forms in the next step.

Function Simplification

Before differentiating a complex function, simplify it. This can be done by rewriting it in a form that’s easier to manage. Our initial function involved radicals and fractions:\[\ f(x) = 12 \sqrt[3]{x^2} + \frac{48}{\sqrt[3]{x}} \]
When simplified, it became:\[ f(x) = 12x^{2/3} + 48x^{-1/3} \]
Why simplify? Here's a few benefits:

• Reduces Complexity: With fewer intricate terms, it’s easier to apply differentiation rules.
• Eliminates Common Mistakes: By converting into basic power or polynomial forms, common calculation errors can be avoided.
• Prepares for Further Steps: Simplification sets up the function for seamless application of rules, like the power rule, as seen in our problem.

This technique is essential, not just for derivatives, but also for calculus in general.