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Chapter 3: Problem 5

Find the derivative. Assume \(a, b, c, k\) are constants. $$y=x^{4 / 3}$$

Short Answer

Expert verified
The derivative is \( y' = \frac{4}{3}x^{1/3} \).

Step by step solution

01

Power Rule Identification

First, identify the function form. The function given is \( y = x^{4/3} \), which is a power function of \( x \). The power rule for derivatives states that if \( y = x^n \), then the derivative \( y' = nx^{n-1} \).
02

Apply the Power Rule

Apply the power rule to find the derivative of the function. The exponent \( n \) in this case is \( \frac{4}{3} \). Thus, the derivative is: \[ \frac{d}{dx}x^{4/3} = \frac{4}{3}x^{4/3 - 1} \].
03

Simplify the Expression

Simplify the expression obtained. The expression \( 4/3 - 1 \) simplifies to \( 1/3 \). Therefore, the derivative of the function is: \[ y' = \frac{4}{3}x^{1/3} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

power rule
The power rule is a fundamental concept in calculus for finding the derivative of a function that is a power of a variable. It provides a quick way to differentiate expressions of the form \(x^n\). - According to the power rule, the derivative of \(y = x^n\) is \(y' = nx^{n-1}\). - The rule applies when \(n\) is any real number, whether positive, negative, or a fraction.
To use the power rule, you simply bring the exponent in front of the variable and then reduce the exponent by one. This simplification greatly helps in dealing with both simple and complex functions. Remember, applying the power rule correctly can save time and reduce errors in calculations.
derivative of power function
Finding the derivative of a power function is straightforward with the power rule in hand. Let's consider the original exercise \(y = x^{4/3}\). - Here, our power function has an exponent of \(\frac{4}{3}\). - Using the power rule, we multiply \(\frac{4}{3}\) by \(x\) raised to the power of \(\frac{4}{3} - 1\).
This gives us the initial derivative as \(\frac{4}{3}x^{1/3}\). This process highlights the importance of the power rule in calculating derivatives of many algebraic expressions with ease. The ability to find derivatives quickly can be powerful in solving differential equations and analyzing mathematical models.
simplifying expressions
Simplifying expressions is an important aspect of mathematics. After applying the power rule, you'll often need to simplify to make it easier to interpret or further use the derivative. - In our example, simplifying involves calculating \(\frac{4}{3} - 1\), which equals \(\frac{1}{3}\).
This reduction simplifies the expression to \(y' = \frac{4}{3}x^{1/3}\). By expressing derivatives in their simplest form, you can more easily integrate them within larger problems and share results in an easily understandable way. Remember, always looking for ways to simplify expressions can help you maintain accuracy and clarity in your mathematical solutions.

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Most popular questions from this chapter

Show that the relative rate of change of a quotient \(f / g\) is the difference between the relative rates of change of \(f\) and \(g\).

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