Problem 21 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, find the third derivative of the function. $$ f(x)=\frac{3}{16 x^{2}} $$

Step-by-Step Solution

Answer

The third derivative of the function \( f(x)=\frac{3}{16 x^{2}} \) is \( f'''(x)=-\frac{9}{2}x^{-5} \).

Step 1: First Derivative

Apply the power rule, which states that the derivative of \(x^n\) where \(n\) is any real number is \(nx^{n-1}\). Thus, for \(f(x)=\frac{3}{16 x^{2}} =\frac{3}{16}*x^{-2}\), the first derivative is \(f'(x)=\frac{3}{16}*(-2)*x^{-2-1} =-\frac{6}{16}x^{-3} = -\frac{3}{8}x^{-3}\).

Step 2: Second Derivative

For the second derivative, apply the power rule again to \(f'(x)= -\frac{3}{8}x^{-3}\), giving \(f''(x)= -\frac{3}{8}*(-3)*x^{-3-1} =\frac{9}{8}x^{-4}\).

Step 3: Third Derivative

Finally, for the third derivative, apply the power rule one more time to \(f''(x)=\frac{9}{8}x^{-4}\), which gives \(f'''(x)= \frac{9}{8}*(-4)*x^{-4-1} =-\frac{36}{8}x^{-5} = -\frac{9}{2}x^{-5}\).

Key Concepts

Power Rule

The power rule is an essential tool in calculus that simplifies the process of finding the derivative of a function involving a term raised to a power, like \(x^n\). The rule states: to differentiate \(x^n\), the result is \(nx^{n-1}\). This rule is extremely useful because it allows you to quickly find the derivative without expanding the term or using more complex differentiation techniques.
For instance, if you have a function where \(f(x) = x^3\), applying the power rule gives \(f'(x) = 3x^{3-1} = 3x^2\).

In the exercise, the function \(f(x) = \frac{3}{16}x^{-2}\) was differentiated into the first derivative using the power rule: \(f'(x) = -\frac{3}{8}x^{-3}\). This straightforward multiplication and exponent adjustment help you quickly derive higher-order derivatives as shown with the second and third derivatives in the problem.

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. It is fundamental for understanding changes and is used across various disciplines such as physics, engineering, economics, and biology.

The differentiation process is a key calculus operation that helps determine how a change in one quantity affects another. Calculus provides tools and formulas, like the power rule, that make solving complex problems more manageable.
In this exercise, we dealt with the process of finding the third derivative, demonstrating one aspect of calculus in action. Calculus aids in uncovering new insights into the behavior and properties of functions, especially those involving rates of change.

Differentiation

Differentiation is the action of calculating a function's derivative, which indicates how the function changes as its input changes. It is a critical concept in calculus and serves as the backbone for numerous mathematical applications.

• Derivatives convey important information about the slope and curvature of a curve.
• Higher-order derivatives, like the third derivative, can indicate the concavity of graphs and more subtle features of the function.

In the exercise example, differentiation allowed the transformation of \(f(x) = \frac{3}{16}x^{-2}\) into its third derivative \(f'''(x) = -\frac{9}{2}x^{-5}\). Each step uses the power rule to methodically reduce the function to its specific derivative order, highlighting how differentiation opens the door to deeper analysis of the function's behavior.