Problem 48 - Calculus an Applied Approach

Question

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

Step-by-Step Solution

Answer

The first derivative of the function \(f(x)=x^{1 / 3}-1\) is \(f^{\prime}(x) = (1/3) \cdot x^{-2/3}\)

Step 1: Identify the correct differentiation rule

For the term \(x^{1 / 3}\), one must apply the power rule for differentiation, which is \((x^n)^{\prime}=n \cdot x^{(n-1)}\). The term \(-1\) is a constant and the derivative of a constant is zero.

Step 2: Apply the power rule

To differentiate \(x^{1/3}\), apply the power rule. From the rule, \(n=1/3\) (as the power), then the derivative would be \((1/3) \cdot x^{(1/3 - 1)} = (1/3) \cdot x^{-2/3}\).

Step 3: Differentiate the Constant

The derivative of a constant is zero. Therefore, the derivative of \(-1\) is 0.

Step 4: Combine the Results

Combine the derivatives of all terms to get the derivative of the entire function, namely: \(f^{\prime}(x)= (1/3) \cdot x^{-2/3} - 0 = (1/3) \cdot x^{-2/3}\).

Key Concepts

Power Rule for Differentiation

Understanding the power rule for differentiation is a cornerstone for grasping basic calculus concepts. Whenever you encounter a function where the variable, often written as 'x', is raised to a power, such as in a polynomial, the power rule will be your method of choice for finding the derivative. The rule itself is straightforward: Given a term in the form of \(x^n\) where 'n' is any real number, its derivative is \(nx^{n-1}\). Let's apply this to a simple term like \(x^{1/3}\). Using the power rule, we arrive at \((1/3)x^{1/3-1}\), which simplifies to \((1/3)x^{-2/3}\).

Why does this work? It's basically a consequence of the limit definition of the derivative. The rule is derived from manipulating the expression that defines the derivative as the limit of the average change in the function as the change in the input approaches zero. This manipulation, involving some clever algebra and limits, yields the power rule's simple formula, saving us from complex limit calculations every time we need to find a derivative.

Derivative of a Constant

Sometimes you'll come across constants in functions, like \(-1\) in the given exercise. A constant is a value that doesn't change; it’s not influenced by the variable 'x' at all. While variables can grow and shrink and do all sorts of things as 'x' changes, constants remain, well, constant! In terms of calculus and differentiation, this means that the derivative of any constant is zero. This rule makes intuitive sense because if you graph a constant value, you'll get a horizontal line, and the slope of a horizontal line—a measure of how steep it is—is naturally zero. By remembering that the derivative represents the rate of change of a function, it’s clear why a constant that doesn’t change at all has a derivative of zero. In our exercise, the derivative of \(-1\) is thus 0, reflecting the fact that it does not change as 'x' varies.

Combining Derivatives

When you're faced with a function composed of several terms like \(x^{1/3} - 1\), knowing how to combine the derivatives of each part is as critical as knowing how to differentiate those parts. After applying the power rule and identifying derivatives of constants, you have to bring the different pieces together. The magic lies in understanding that the derivative of a sum (or difference) of functions is simply the sum (or difference) of their respective derivatives. This additive property allows us to tackle each term individually and then just sum up the results.

In the exercise at hand, we combined the derivative of \(x^{1/3}\), which is \((1/3) \times x^{-2/3}\), with the derivative of the constant term \(-1\), which is 0. There's no change from the constant, so the combined derivative of the function \(f(x) = x^{1/3} - 1\) is just \(f'(x) = (1/3) \times x^{-2/3}\), neat and tidy. Remember, when combining derivatives, just add or subtract the derivatives of the individual terms, and you've got your answer!