Problem 4 - Calculus Single Variable

Question

An expression for \(f(x)\) is given. Compute the first, second, and third derivatives of \(f(x)\) with respect to \(x\). \(x^{7 / 3}\)

Step-by-Step Solution

Answer

The first derivative is \(\frac{7}{3}x^{4/3}\), the second is \(\frac{28}{9}x^{1/3}\), and the third is \(\frac{28}{27}x^{-2/3}\).

Step 1: Find the first derivative

To find the first derivative, we apply the power rule, which states that if \(f(x) = x^{n}\), then \(f'(x) = nx^{n-1}\). For \(f(x) = x^{7/3}\), the first derivative is \(f'(x) = \frac{7}{3}x^{\frac{7}{3} - 1} = \frac{7}{3}x^{\frac{4}{3}}\).

Step 2: Find the second derivative

For the second derivative, we again apply the power rule to \(f'(x) = \frac{7}{3}x^{4/3}\). The second derivative is \(f''(x) = \frac{7}{3} \cdot \frac{4}{3}x^{\frac{4}{3} - 1} = \frac{28}{9}x^{\frac{1}{3}}\).

Step 3: Find the third derivative

We continue by applying the power rule to \(f''(x) = \frac{28}{9}x^{1/3}\). The third derivative, \(f'''(x)\), is \(f'''(x) = \frac{28}{9} \cdot \frac{1}{3}x^{\frac{1}{3} - 1} = \frac{28}{27}x^{-\frac{2}{3}}\).

Key Concepts

Power Rule

The power rule is a fundamental part of calculus, allowing us to find derivatives quickly and efficiently. It's particularly useful when dealing with functions of the form \( x^n \). The rule tells us that the derivative of \( x^n \) is \( nx^{n-1} \). This means you take the exponent, bring it down as a coefficient, and subtract one from the original exponent. This simple rule helps in calculating derivatives of polynomial functions, making complex calculus problems more manageable.

For instance, if you have a function like \( f(x) = x^{7/3} \), applying the power rule means taking \( 7/3 \) as a coefficient and reducing the index by one, resulting in \( f'(x) = \frac{7}{3}x^{4/3} \). The power rule provides a clear path to differentiate polynomials efficiently.

First Derivative

The first derivative of a function provides crucial information about its rate of change. It is the initial step in understanding how a function behaves. By using the power rule, you can quickly find the first derivative. For our function \( f(x) = x^{7/3} \), the first derivative is \( f'(x) = \frac{7}{3}x^{4/3} \).

This derivative tells us the slope of the tangent line to the curve at any point \( x \). A positive value indicates the function increases, whereas a negative value indicates it decreases. Understanding the first derivative is essential for analyzing graphs and determining maxima, minima, or inflection points.

Second Derivative

The second derivative represents the derivative of the first derivative, offering insight into the concavity of the function and the acceleration of its rate of change. For the function \( f'(x) = \frac{7}{3}x^{4/3} \), using the power rule again gives us the second derivative as \( f''(x) = \frac{28}{9}x^{1/3} \).

Knowing the concavity helps determine if a graph is bending upwards or downwards at certain points. If the second derivative is positive, the graph is concave up, resembling a bowl shape, while a negative second derivative indicates a concave down shape. This is crucial in locating points of inflection where the graph changes curvature.

Third Derivative

The third derivative is less commonly discussed than the first and second, but it provides higher-order insights into the function's behavior. It is the rate of change of the second derivative, denoting how the curvature of the function itself changes over time. For \( f''(x) = \frac{28}{9}x^{1/3} \), we apply the power rule for the third time to find \( f'''(x) = \frac{28}{27}x^{-2/3} \).

Though rarely necessary for basic curve analysis, the third derivative can be valuable in advanced applications, such as studying motion in physics or complex modeling. High-order derivatives help in understanding finer details of the function's structure and behavior over different intervals.