Problem 15 - Brief Applied Calculus

Question

Find the second derivative of each function. $$ f(x)=\frac{27}{\sqrt[3]{x}} $$

Step-by-Step Solution

Answer

The second derivative is \( f''(x) = \frac{12}{x^{\frac{7}{3}}} \).

Step 1: Rewrite the Function

Start by rewriting the function with a power notation, as this will make it easier to differentiate. The function is given as \( f(x) = \frac{27}{\sqrt[3]{x}} \). This can be rewritten using exponents: \( f(x) = 27x^{-\frac{1}{3}} \).

Step 2: Find the First Derivative

Differentiate the function \( f(x) = 27x^{-\frac{1}{3}} \) using the power rule \( \frac{d}{dx}[x^n] = nx^{n-1} \). Thus, the first derivative \( f'(x) \) is \( f'(x) = 27 \cdot (-\frac{1}{3}) x^{-\frac{1}{3} - 1} = -9x^{-rac{4}{3}} \).

Step 3: Find the Second Derivative

Differentiate the first derivative \( f'(x) = -9x^{-\frac{4}{3}} \) again using the power rule. The second derivative \( f''(x) \) is given by \( f''(x) = -9 \cdot (-\frac{4}{3}) x^{-\frac{4}{3} - 1} = 12x^{-\frac{7}{3}} \).

Step 4: Simplify the Second Derivative

Rewrite the second derivative \( f''(x) = 12 x^{-\frac{7}{3}} \) to its original radical form, if necessary. This is \( f''(x) = \frac{12}{x^{\frac{7}{3}}} \).

Key Concepts

Power Rule

The power rule is a fundamental tool in calculus used for differentiating functions involving power terms. It's particularly handy for differentiating expressions where variables are raised to constant powers. The rule is straightforward: to differentiate a function of the form \( x^n \), multiply by the power \( n \) and reduce the power by one, i.e., \( \frac{d}{dx}[x^n] = nx^{n-1} \). This effectively reduces the power of the variable by one degree, while maintaining the same coefficient—a simple yet powerful technique.

In the given exercise, applying the power rule allows the transformation of the original function, \( f(x) = \frac{27}{\sqrt[3]{x}} \), into a format more amenable to differentiation. It turns into \( 27x^{-\frac{1}{3}} \) which is now a straightforward candidate for the power rule. This step simplifies the problem significantly by converting roots and denominators to exponents. Thus, making the differentiation process logical and systematic.

First Derivative

The first derivative of a function provides us with the rate at which the function's output changes with respect to change in input. It's essential for understanding the function's behavior, such as its slope and tendency—whether it's increasing or decreasing. In this context, the differentiation of the rewritten function yields useful insights.

For the function \( f(x) = 27x^{-\frac{1}{3}} \), employing the power rule results in the first derivative: \( f'(x) = -9x^{-\frac{4}{3}} \). Here's what the process looks like with the power rule applied:

• The original exponent \(-\frac{1}{3}\) is reduced by 1 to get \(-\frac{4}{3}\).
• Multiply the coefficient (27) by the new exponent \(-\frac{1}{3}\) to get \(-9\).

The first derivative, \( f'(x) \), provides a new function that can inform on the instantaneous rate of change at any point \( x \) on the original function curve. Reviewing how functions change with derivatives helps build an understanding of their increasing and decreasing intervals.

Second Derivative

The second derivative of a function tells us about the concavity of the function and its acceleration; it is a derivative of the first derivative. In essence, it answers questions about how the rate of change is changing, shedding light on the curvature of the function.

For our exercise, the first derivative \( f'(x) = -9x^{-\frac{4}{3}} \) is differentiated using the power rule once more to obtain the second derivative, \( f''(x) = 12x^{-\frac{7}{3}} \). The calculation involves:

• Multiplying \(-9\) by \(-\frac{4}{3}\), leading to the coefficient 12.
• Subtracting 1 from the exponent \(-\frac{4}{3}\) to get \(-\frac{7}{3}\).

This process iteratively applies the power rule, and the resulting second derivative offers valuable information on the function's concavity. If \( f''(x) > 0 \), the function is concave up and vice versa if \( f''(x) < 0 \). Thus, the second derivative is crucial in understanding how the graph of a function bends.

Function Differentiation

Function differentiation is a core process in calculus that involves finding how a function changes at any given point. Differentiation transforms a function into its derivative, providing critical information such as slope, velocity, and acceleration, all of which are vital for analyzing real-world phenomena.

When differentiating functions like \( f(x) = \frac{27}{\sqrt[3]{x}} \), the initial task often involves reformatting the function. By converting division and roots into exponents, as seen here with \( 27x^{-\frac{1}{3}} \), we facilitate easier differentiation practices.

This differentiation process unfolds in multiple steps for different orders, each providing deeper insights into the function:

• The **first derivative** gives us the gradient or slope; it answers how steep the curve is at any point.
• The **second derivative** informs on concavity and informs about the acceleration of change.

Each derivative peels back another layer, allowing us to predict and understand the behavior of functions in increasingly sophisticated ways. Function differentiation stands as a pivotal aspect of calculus, underpinning much of its utility and application.