Problem 34 - Calculus an Applied Approach

Question

Use the General Power Rule to find the derivative of the function. $$ y=\sqrt[3]{3 x^{3}+4 x} $$

Step-by-Step Solution

Answer

The derivative of the function \(y = \sqrt[3]{3x^3 + 4x}\) is \(y' = (1/3)*(3x^3 + 4x)^{-2/3}*(9x^2 + 4)\).

Step 1: Rewrite the Function

Rewrite the function as \(y = (3x^3 + 4x)^{1/3}\) to make it easier to apply the general power rule.

Step 2: Find the Derivative of the Base

Take the derivative of the base, \(u = 3x^3 + 4x\). The derivative \(u'\) will be \(u' = 9x^2 + 4\).

Step 3: Apply the General Power Rule

The General Power Rule is \((u^n)' = n*u^{n-1}*u'\). Here, \(n = 1/3\), \(u = 3x^3 + 4x\) and \(u' = 9x^2 + 4\). Substituting these values into the power rule gives us \(y' = (1/3)*(3x^3 + 4x)^{-2/3}*(9x^2 + 4)\).

Key Concepts

Understanding Derivatives

Derivatives are a core concept in calculus, representing how a function changes as its input changes. They provide information about the slope of the function at any given point. Imagine you're driving a car; the speed of the car is like the derivative of your position with respect to time.

In calculus, the derivative is often expressed as \( f'(x) \) or \( \frac{df}{dx} \). It helps to understand how quantities change and interact. Let's keep in mind:

• The derivative of a constant is zero.
• The derivative of a sum is the sum of the derivatives.
• A derivative can tell us where a function increases or decreases.

To solve practical problems, derivatives are used to find things like maxima and minima, which often represent the best or worst-case scenarios in scenarios such as profit maximization or minimizing costs.

Exploring Power Functions

Power functions are a type of mathematical function characterized by the variable raised to a power. They can be written in the form \( y = ax^n \), where \( a \) and \( n \) are constants. These functions exhibit a simple yet important behavior that can be analyzed using calculus tools like derivatives.

When dealing with power functions, specific rules apply, such as:

• The derivative of \( x^n \) is given by the formula \( nx^{n-1} \).
• A power function \((x^n)' \) changes based on \( n \), affecting speed and direction.

In our exercise, the function \( y = (3x^3 + 4x)^{1/3} \) is a composite function where the power rule is used to find its derivative. Recognizing and handling such functions is key to efficiently solving calculus problems.

Approaches to Calculus Problem Solving

The step-by-step method to solving calculus problems often involves recognizing the function type and applying appropriate rules like the General Power Rule. This involves:

• Identifying the function's structure.
• Rewriting the function in a simpler form, if necessary.
• Calculating the derivative of each component.
• Applying derivative rules efficiently to solve the problem.

In our original exercise, we start by rewriting the function to better apply the General Power Rule. This process involves breaking down a complex composition into manageable parts. By isolating parts of the function, you can individually find the derivatives and combine them using calculus rules.

This structured approach makes even challenging calculus problems more manageable and lays the groundwork for handling a wide variety of calculus-based tasks.