Problem 48 - College Algebra and Calculus: An Applied Approach

Question

find \(f^{\prime}(x)\). $$ f(x)=x^{1 / 3}-1 $$

Step-by-Step Solution

Answer

\(f^\prime(x) = (1/3)x^{-2/3}\)

Step 1: Recall the Power Rule

Recall the power rule for derivatives: \(f(x) = x^n => f^\prime(x) = nx^{n-1}\)

Step 2: Apply the Power Rule

The function has two terms: \(x^{1/3}\) and -1. First, to differentiate \(x^{1/3}\), using the power rule, the derivative would be \((1/3)x^{1/3 - 1} = (1/3)x^{-2/3}\). Next, the derivative of a constant (-1) is 0.

Step 3: Write the Final Answer

The derivatives of each term add up to form the final derivative of the function. Therefore, \(f^\prime(x) = (1/3)x^{-2/3} - 0 = (1/3)x^{-2/3}\).

Key Concepts

Power Rule for Derivatives

One of the most fundamental concepts in calculus is the power rule for finding derivatives. The power rule gives a quick and simple method for differentiating functions of the form f(x) = x^n, where n is any real number. According to the power rule, the derivative of such a function is f'(x) = nx^{n-1}. In essence, to differentiate a power function, you multiply the original exponent n by the base x, then subtract one from the exponent.

For example, if we apply the power rule to f(x) = x^5, we multiply 5 by the base x and then decrease the exponent by one to get f'(x) = 5x^4. This rule applies to both positive and negative exponents, making it very versatile for various calculus problems. Therefore, having a strong grasp of the power rule is essential when taking derivatives in calculus.

Derivative of Constants

It's also crucial to understand how to handle constants when taking derivatives. A constant is a fixed value that does not change, such as the number 5 or -3. In calculus, the derivative of any constant is zero. This is because a constant does not change, and the derivative measures the rate of change of a function.

For a function like f(x) = 7, where 7 is a constant, the derivative f'(x) is 0 because there is no change in value as x changes. The concept is simple but important as it often simplifies the process of differentiating more complex functions. When combined with the power rule, this makes the job of finding derivatives of polynomial functions relatively straightforward.

Applying the Power Rule

Applying the power rule in practice involves identifying the power functions in the expression and differentiating term by term. We have already seen that the power rule involves bringing down the exponent as a multiplier and then subtracting one from the exponent.

Looking back at the exercise f(x) = x^{1/3} - 1, we can apply the power rule to x^{1/3} and the derivative of constants to the -1. The application results in f'(x) = (1/3)x^{-2/3}, since (1/3) comes down as a coefficient and the new exponent becomes -2/3 when we subtract one. The constant term's derivative is zero and thus does not appear in the final answer. It is important to follow each step accurately and also to simplify the final expression if possible.