Problem 16 - Brief Applied Calculus

Question

Find the second derivative of each function. $$ f(x)=\frac{32}{\sqrt[4]{x}} $$

Step-by-Step Solution

Answer

The second derivative is \( f''(x) = 10x^{-9/4} \).

Step 1: Rewrite the Function

First, rewrite the function \( f(x) = \frac{32}{\sqrt[4]{x}} \) using exponents instead of roots. Recall that \( \sqrt[4]{x} = x^{1/4} \), so the function can be rewritten as \( f(x) = 32x^{-1/4} \).

Step 2: Compute the First Derivative

Apply the power rule to find the first derivative of \( f(x) = 32x^{-1/4} \). The power rule states that \( \frac{d}{dx} [x^n] = nx^{n-1} \). For \( f(x) = 32x^{-1/4} \), the first derivative is \( f'(x) = 32 \cdot (-1/4)x^{-5/4} = -8x^{-5/4} \).

Step 3: Compute the Second Derivative

Apply the power rule again to the first derivative \( f'(x) = -8x^{-5/4} \). The derivative \( f''(x) \) is obtained as \( f''(x) = -8 \cdot (-5/4)x^{-9/4} = 10x^{-9/4} \).

Step 4: Simplify the Expression

Rewrite the second derivative \( f''(x) = 10x^{-9/4} \) as a fraction with roots: \( f''(x) = \frac{10}{x^{9/4}} \), if more familiar.

Key Concepts

Understanding the Power Rule

In calculus, the power rule is an essential tool for finding derivatives, especially for functions involving power terms. The power rule states that if you have a function of the form \( f(x) = x^n \), where \( n \) is any real number, then the derivative is given by \( f'(x) = nx^{n-1} \).
This essentially means you multiply the original exponent by the coefficient (if any) and then subtract one from the exponent. It's a quick and straightforward method used widely for differentiating polynomial functions.For example, to differentiate \( f(x) = 32x^{-1/4} \), as in the original problem, you multiply \( -1/4 \) by 32 to get \( -8 \), then decrease the exponent by one, resulting in \( -5/4 \).

• Original Function: \( f(x) = 32x^{-1/4} \)
• First Derivative: Use power rule to get \( f'(x) = -8x^{-5/4} \)

Understanding this rule is vital for solving more complicated calculus problems, as you'll often need to apply it multiple times, especially when finding higher-order derivatives like the second or third derivative.

The Role of Exponents

Exponents play a crucial role in calculus problem-solving, particularly when dealing with derivatives. In mathematics, an exponent indicates how many times a number, known as the base, is multiplied by itself.
In the function \( f(x) = 32x^{-1/4} \), the exponent \(-1/4\) suggests the reciprocal of the fourth root of \( x \). When calculating derivatives, changing the exponent is a natural step from the application of the power rule.A change in the exponent can significantly alter the function's behavior, influencing the slope, curve, and overall shape of the graph.

• Conversion: Convert roots into exponents for easier calculation.
• Simplification: Exponents help in simplifying the derivative process.

In the given problem, transitioning from the power of \(-1/4\) in the original function to \(-5/4\) in the first derivative illustrates how differentiation affects exponents systematically.
This exponent manipulation continues as you proceed to the second derivative, thereby highlighting its ever-present influence.

Strategies for Calculus Problem Solving

Solving calculus problems, especially those involving derivatives, often requires a systematic approach. Knowing rules and best practices can simplify the process, minimizing error and maximizing efficiency. Here’s a broad strategy:- **Conversion to Simpler Forms**: Rewrite complex terms using simpler equivalents. For example, rewriting roots as exponents, like \( \sqrt[4]{x} = x^{1/4} \), helps in applying derivative rules efficiently.- **Apply the Right Rule**: Once simplified, select the most suitable calculus rule, like the power rule, to progress.- **Derivative Chains**: When calculating higher-order derivatives, take the derivative step-by-step. Start from the first and proceed to the second, using the logic and formula provided in each step.
Reapplying steps with precision is essential. As seen in the original problem,

• First, the function is simplified and the first derivative is found using the power rule.
• Secondly, the second derivative is calculated, following the same principle.

Remember, practice and thorough understanding of these strategies can significantly improve your calculus problem-solving skills. With regular use, these methods become second nature, allowing you to tackle even more challenging calculus tasks with confidence.