Problem 16 - Calculus and its applications

Question

Find \(f^{\prime \prime}(x)\) $$ f(x)=x^{1 / 3} $$

Step-by-Step Solution

Answer

The second derivative is \(f''(x) = -\frac{2}{9}x^{-5/3}\) or \(f''(x) = -\frac{2}{9x^{5/3}}\).

Step 1: Find the First Derivative of f(x)

To find the first derivative of the function \(f(x) = x^{1/3}\), we apply the power rule for derivatives, which states that \(\frac{d}{dx}(x^n) = nx^{n-1}\). Here, \(n = \frac{1}{3}\), thus \(f'(x) = \frac{1}{3}x^{1/3 - 1} = \frac{1}{3}x^{-2/3}\).

Step 2: Find the Second Derivative of f(x)

Now, calculate the second derivative by differentiating \(f'(x) = \frac{1}{3}x^{-2/3}\) again using the power rule. We have \(n = -\frac{2}{3}\), so \(f''(x) = \frac{1}{3}\left(-\frac{2}{3}\right)x^{-2/3 - 1} = -\frac{2}{9}x^{-5/3}\).

Step 3: Simplify the Second Derivative

Finally, express the second derivative in a simplified form. From \(f''(x) = -\frac{2}{9}x^{-5/3}\), it can also be written as: \(f''(x) = -\frac{2}{9x^{5/3}}\).

Key Concepts

Power Rule

The power rule is a fundamental concept in differentiation, helping us easily find derivatives of functions with powers of variables. It states that if you have a function in the form of \(x^n\), the derivative is \(nx^{n-1}\). This rule simplifies the process of differentiation greatly, converting complex polynomials into manageable forms. For any real number exponent \(n\), simply drop the exponent as a coefficient and reduce the power by one.
For example, if you start with \(f(x) = x^{1/3}\), you apply the power rule where \(n = 1/3\), leading to \(f'(x) = \frac{1}{3}x^{-2/3}\). This same technique is also applied when moving from the first derivative to the second derivative.

Differentiation

Differentiation is a key operation in calculus used to determine how a function changes as its input changes. When you differentiate a function, you are looking at the rate of change or the slope of the function at any given point. It's a powerful tool used in various fields such as physics, engineering, and economics.
When differentiating, applying rules like the power rule helps streamline the process. In our exercise, finding the first and then the second derivative of \(f(x) = x^{1/3}\) required using the power rule twice. Initially, you derive \(f'(x)\), and subsequently differentiate again to find \(f''(x)\). This process demonstrates how differentiation can be applied multiple times on a function.

Calculus Problem-Solving

Tackling calculus problems often involves recognizing which methods and rules apply to the given function. Successful problem-solving requires:

• Understanding the function's form and behavior.
• Applying appropriate differentiation rules, like the power rule.
• Simplifying the results into workable forms.

In our specific example, through methodical steps, we first found the first derivative and then progressed to the second derivative. Recognizing operational steps, like using the power rule, ensures consistent problem-solving strategies in calculus. Remember, the practice of breaking down the problems into smaller, manageable parts can significantly aid in understanding and ultimately mastering calculus.