Problem 22 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, find the third derivative of the function. $$ f(x)=\frac{1}{x} $$

Step-by-Step Solution

Answer

The third derivative of the function is \( f'''(x) = -\frac{6}{x^4} \).

Step 1 - Find the first derivative

Rewrite the function \( f(x) = \frac{1}{x} \) as \( f(x) = x^{-1} \). Now, apply the power rule on this function to find the first derivative, using the rule \( (x^n)' = n*x^{n-1} \). The first derivative \( f'(x) = -1*x^{-2} = -\frac{1}{x^2} \).

Step 2 - Find the second derivative

Next, find the second derivative by applying the power rule again to the first derivative \( f'(x) = -\frac{1}{x^2} \), rewritten as \( f'(x) = -x^{-2} \). The second derivative \( f''(x) = 2*x^{-3} = \frac{2}{x^3} \).

Step 3 - Find the third derivative

Lastly, apply the power rule once more to the second derivative \( f''(x) = \frac{2}{x^3} \), rewritten as \( f''(x) = 2x^{-3} \). The third derivative \( f'''(x) = -6*x^{-4} = -\frac{6}{x^4} \).

Key Concepts

Power Rule in Calculus

The power rule is a fundamental concept in calculus used to find the derivative of a power function. A power function is one where the variable is raised to a constant power, like \( x^n \). The rule states that to differentiate \( x^n \), you can multiply by the power \( n \) and then subtract one from the exponent. So, the derivative is \( n*x^{n-1} \).

It's useful because it makes differentiation straightforward for power functions. For example, if you have \( x^3 \), the derivative using the power rule is \( 3*x^{2} \). It's not restricted to positive integers; it can be applied to any real number, including negative and fractional exponents.

To illustrate: if \( n \) is -1, as in the exercise \( f(x) = x^{-1} \), applying the power rule gives us \( -1*x^{-2} \). This simplicity and flexibility make the power rule a handy tool for calculus students.

Understanding Derivatives

A derivative represents the rate of change of a function concerning its variable. In simpler terms, it tells us how the output of a function changes when we slightly change the input.

Derivatives are widely used to understand the behavior of functions, such as finding slope, velocity, and in this case, higher-order derivatives like the third derivative. To find a derivative, we use differentiation techniques like the power rule. With functions like \( f(x) = \frac{1}{x} \), derivatives can help us find patterns in how the function behaves.

Taking higher-order derivatives, like a third derivative, means applying the differentiation process multiple times. As shown with \( f(x) = \frac{1}{x} \), the first derivative, second derivative, and third derivative become progressively different in form: \( -\frac{1}{x^2} \), \( \frac{2}{x^3} \), and \( -\frac{6}{x^4} \), respectively. These derivatives reveal more about the function's curvature and rate of change over time.

Steps in Calculus for Differentiation

Differentiation involves a set of steps to methodically find derivatives. It's essential to rewrite functions in power form if needed, as seen in the exercise with \( f(x) = \frac{1}{x} \). This is rewritten as \( x^{-1} \) to apply the power rule more easily.

Here's a simplified process to follow:

• Rewrite the Function: Ensure the function expresses the variable in a polynomial form (e.g., \( x^{-1} \)).
• Apply the Power Rule: Use the power rule \( n*x^{n-1} \) for each step, finding the first, second, or nth derivative.
• Simplify Each Result: After applying the power rule, simplify expressions where possible, like rewriting \( -1*x^{-2} \) as \( -\frac{1}{x^2} \).

This structured approach helps tackle even complex functions by breaking them down into manageable steps. Each differentiation step follows logically from the previous one, ensuring clarity and precision in solving calculus problems.