Problem 2 - Calculus

Question

Find \(D_{x} y\) using the rules of this section. $$y=3 x^{3}$$

Step-by-Step Solution

Answer

The derivative is \( D_{x} y = 9x^2 \).

Step 1: Identify the Function

The function given is in the form of a power function: \( y = 3x^3 \). This means we have a function where \( x \) is raised to a power.

Step 2: Recall the Power Rule for Differentiation

The power rule states that if \( y = ax^n \), then \( \frac{dy}{dx} = anx^{n-1} \). In this case, the exponent \( n \) is \( 3 \), and the coefficient \( a \) is \( 3 \) as well.

Step 3: Apply the Power Rule

To find \( D_{x} y \), apply the power rule: \( \frac{dy}{dx} = 3 \times 3x^{3-1} = 9x^2 \).

Step 4: Final Answer

The derivative of the function is \( D_{x} y = 9x^2 \).

Key Concepts

Power Rule

The Power Rule is a simple yet powerful tool in calculus, especially when dealing with polynomial functions. This rule helps us find the derivative of a function where a variable is raised to a power. In general, if you have a function of the form \( y = ax^n \), the Power Rule allows you to easily find its derivative.
Here is how it works:

• Multiply the exponent \( n \) by the coefficient \( a \) of the variable.
• Decrease the exponent \( n \) by one.

By applying these steps, you get the derivative: \( \frac{dy}{dx} = anx^{n-1} \).
This means you are finding the rate of change of \( y \) with respect to \( x \). The Power Rule simplifies the process of differentiation, making it straightforward to handle polynomial expressions.

Derivative Calculation

Calculating the derivative of a function is an essential process in calculus, and the Power Rule simplifies derivative calculations significantly for any polynomial-type function.
For example, let's revisit the function \( y = 3x^3 \). According to the Power Rule, in which the function is in the form \( y = ax^n \), there are specific steps to follow:

• Identify the coefficient \( a \) as \( 3 \).
• Notice the exponent \( n \), which is \( 3 \).
• Apply the formula by multiplying these parts: \( 3 \times 3 \), resulting in \( 9 \).
• Reduce the exponent by one, leading to a new power of \( x^2 \).

Thus, the derivative \( \frac{dy}{dx} \) becomes \( 9x^2 \).
This result shows the rate at which the original function \( y \) changes concerning \( x \). Understanding how to perform these calculations can significantly ease the learning of complex calculus concepts.

Function Identification

Before starting any differentiation, identifying the function type is crucial. This helps determine the best method for finding the derivative. In this case, the function given is \( y = 3x^3 \), which is a power function.
A power function typically features:

• A constant multiplied by a variable raised to an exponent.
• The general form \( y = ax^n \), where \( a \) and \( n \) are constants, and \( x \) is a variable.

Properly recognizing this form allows us to apply the Power Rule, thereby simplifying the differentiation process.
Correct identification ensures you utilize the appropriate rules for calculus, reducing errors, and improving understanding of how functions behave. This foundational step is vital in tackling more complex mathematical problems effectively.