Problem 65 - Calculus and its applications

Question

Use a calculator to determine where \(f^{\prime}(x)\) does not exist, if \(f(x)=\sqrt[3]{x-5}\).

Step-by-Step Solution

Answer

The derivative \(f^{\prime}(x)\) does not exist at \(x = 5\).

Step 1: Identify the Domain of the Function

The function given is \(f(x) = \sqrt[3]{x-5}\). A cube root function is defined for all real numbers because you can take the cube root of any real number. Hence, the domain of \(f(x)\) is all real numbers.

Step 2: Differentiate the Function

The derivative of the cube root function can be found by applying the power rule. Rewrite the function as \(f(x) = (x-5)^{1/3}\). Using the power rule, \(f^{\prime}(x) = \frac{1}{3}(x-5)^{-2/3} = \frac{1}{3}\cdot \frac{1}{(x-5)^{2/3}}\).

Step 3: Analyze the Derivative for Undefined Points

The derivative \(f^{\prime}(x) = \frac{1}{3(x-5)^{2/3}}\) is undefined where the denominator is zero. For this to happen, \((x-5)^{2/3} = 0\). This occurs at \(x = 5\).

Step 4: Determine Where the Derivative Does Not Exist

Since the derivative is undefined at \(x = 5\), \(f^{\prime}(x)\) does not exist at exactly this point. Check your calculation using a calculator if needed. For cube root functions like this, ensure the problem of the undefined derivative makes intuitive sense due to division by zero.

Key Concepts

Derivative of Cube Root Function

When working with functions that involve roots, like the cube root function, one often needs to understand how to find their derivatives. The cube root function can be expressed in a more calculus-friendly form: \( f(x) = \sqrt[3]{x-5} \) becomes \( f(x) = (x-5)^{1/3} \). This representation allows us to use differentiation rules more effectively.

To find the derivative of the cube root function, we need to recognize that we're dealing with a power function. The derivative of a function gives us the rate at which the function's value is changing concerning its input or variable (in this case, \( x \)). For cube root functions, using the power form makes applying differentiation rules straightforward and helps us identify points where the function's slope might become undefined or behave peculiarly.

Ultimately, the differentiation of a cube root prepares us for further analysis of the function's behavior, including where it might not exist or be undefined, which is crucial for understanding "Undefined Derivative Points" in calculus.

Power Rule in Calculus

The power rule is one of the first and most essential rules that students learn while studying calculus, especially when working with polynomial and root functions. It simplifies the process of finding derivatives and is applicable when dealing with functions expressed as powers of \( x \).

Specifically, the power rule states that if you have a function \( f(x) = x^n \), the derivative \( f^{\prime}(x) \) is \( nx^{n-1} \). For our cube root function, \( (x-5)^{1/3} \), applying the power rule gives us the formula for its derivative: \( \frac{1}{3}(x-5)^{-2/3} \). This shows that for any power expressed as a fraction, we simultaneously lower the power by one and multiply by the original exponent.

This straightforward rule is invaluable because it allows students to tackle more complex, real-world problems, providing insight into the rate of change for functions that appear regularly in mathematics and sciences.

Undefined Derivative Points

In the context of differentiability, understanding where a derivative is undefined is crucial. These points are where the rules of calculus tell us the function lacks a slope because of discontinuities, sharp turns, or vertical tangents.

For the cube root function discussed, \( f^{\prime}(x) = \frac{1}{3(x-5)^{2/3}} \), the derivative becomes undefined when the denominator equals zero. Consequently, solving \((x-5)^{2/3} = 0\) gives us \( x = 5 \). At this point, the derivative tells us the function's rate of change is undefined. Here, the cube root function's behavior is akin to a sharp corner or cusp, which visually and mathematically denotes an undefined derivative.

Recognizing undefined derivative points is not only about solving equations. It's also about gaining insight into the behavior of functions at certain values, which can lead to a deeper understanding of function dynamics in calculus.