Question

Using the Constant Multiple Rule In Exercises 19-24, complete the table to find the derivative of the function without using the Quotient Rule. $$ \text {Function} \quad \text {Rewrite} \quad \text {Differentiate} \quad \text {Simplify} $$ \(y=\frac{2 x}{x^{1 / 3}}\)

Short answer

The derivative of the function \(y=\frac{2 x}{x^{1 / 3}}\) is \(y'=2 x^{2 / 3}-\frac{2}{3} x^{-1 / 3}\).

Step-by-step solution

Rewriting the function

Rewrite the function from \(y=\frac{2 x}{x^{1 / 3}}\) to a form without a quotient. This can be done by recognizing that \(x^{1 / 3}\) is the same as the cube root of \(x\), which can be placed in the denominator of the fraction. Rewriting gives \(y=2 x\cdot x^{-1 / 3}\).

Applying Constant Multiplication Rule

According to the Constant Multiplication Rule, when a function \(f(x)\) is multiplied by a constant \(c\), the derivative of the product is just the constant times the derivative of the function. So, differentiating gives \(y'=2 x^{1-1/3} + 2 x^{2/3}(-1/3) x^{-1}\).

Simplifying

Combine like terms and simplify the expressions to obtain the final derivative. This gives \(y'=2 x^{2 / 3}-\frac{2}{3} x^{-1 / 3}\).

Key concepts

Derivative Without Quotient Rule

Understanding how to compute derivatives without the Quotient Rule is an essential skill in calculus. The Quotient Rule is typically used when differentiating functions that are expressed as one function divided by another. However, in some cases, especially when the functions are power functions, it is more straightforward to rewrite the function into a form that does not require the Quotient Rule.

In the given exercise, we encounter a function that initially appears to require the Quotient Rule for differentiation: \( y = \frac{2x}{x^{1/3}} \). The Quotient Rule could be used here, but it invites unnecessary complexity. Instead, the function is rewritten using properties of exponents. Multiplication and division of exponents can be translated into addition and subtraction of their powers when they have the same base, simplifying the differentiating process.

For instance, when we rewrite \( y = \frac{2x}{x^{1/3}} \) as \( y = 2x \times x^{-1/3} \), we're using the property that \( \frac{1}{x^{a}} = x^{-a} \). This rewrite enables us to apply the Constant Multiple Rule directly, which states that the derivative of a constant times a function is the constant times the derivative of the function. By eliminating the quotient, we avoid potential errors and make the differentiation process more straightforward.

Differentiating Power Functions

When differentiating power functions, we can apply straightforward rules that make the calculation of derivatives much more manageable. A power function is of the form \( f(x) = x^n \), where \( n \) is a constant. The derivative of a power function can be found using the Power Rule, which states that if \( f(x) = x^n \), then \( f'(x) = n \times x^{n-1} \).

In our exercise, after rewriting the function as a power function with a constant multiple, it became \( y = 2x^{1 - \frac{1}{3}} \). Applying the Power Rule to differentiate, we find the derivative of each term separately before combining them. This rule simplifies the process significantly because it allows us to work with each term individually rather than as part of a more complex quotient. As with any rule, ensuring the correct application is critical – remember to subtract one from the exponent and multiply the result by the original exponent value.

Simplifying Derivatives

Simplifying derivatives is an essential step in the differentiation process that helps us to understand and interpret the results better. After applying differentiation rules such as the Power Rule and the Constant Multiple Rule, we often end up with an expression that can be made more concise.

In the solution, we obtain an expression with multiple terms. Simplification in this context may involve combining like terms, reducing fractions, and applying algebraic rules to achieve the simplest form of the derivative. In our example, after differentiating, we arrive at \( y' = 2x^{2/3} - \frac{2}{3}x^{-1/3} \), which is already fairly simple. Additional simplification may not be necessary unless the context of a problem requires it.

Keeping expressions in their simplest form ensures clarity and reduces the likelihood of mistakes in further calculations. It is a good practice to always question whether the derivative can be simplified further, as it promotes a thorough understanding of the relationships between the different parts of the function.