Problem 7 - University Calculus: Early Transcendentals

Question

Find the indicated derivatives. $$\frac{d y}{d x}\( if \)y=2 x^{3}$$

Step-by-Step Solution

Answer

The derivative is \( \frac{dy}{dx} = 6x^2 \).

Step 1: Identify the Derivative Rule

The function given is a polynomial function, \( y = 2x^3 \). For polynomial functions of the form \( ax^n \), the derivative can be found using the power rule. The power rule states that if \( y = ax^n \), then \( \frac{dy}{dx} = nax^{n-1} \).

Step 2: Apply the Power Rule

Apply the power rule to the function \( y = 2x^3 \). Here, \( a = 2 \), \( n = 3 \). According to the power rule, differentiate \( 2x^3 \) with respect to \( x \):\[ \frac{dy}{dx} = 3 \cdot 2 \cdot x^{3-1} = 6x^2 \]

Step 3: Write the Final Derivative

The derivative of the given function \( y = 2x^3 \) is \( \frac{dy}{dx} = 6x^2 \).

Key Concepts

Power Rule

The power rule is a key technique in calculus used to find derivatives, particularly for polynomial functions. It's a simple formula used to differentiate expressions of the form \( ax^n \), where \( a \) is a constant and \( n \) is a positive integer or zero. According to the power rule:

• If \( y = ax^n \)
• Then the derivative \( \frac{dy}{dx} = nax^{n-1} \)

This formula helps in converting the exponent \( n \) into a coefficient \( n \cdot a \) and reducing the power of \( x \) by one.
For example, if we have a function \( y = 2x^3 \), using the power rule, the derivative \( \frac{dy}{dx} \) becomes \( 6x^2 \). This means we've multiplied the original exponent by the coefficient and then decreased the exponent by one.
The power rule is especially useful for functions with multiple terms, as you can apply it to each term individually and combine the results.

Polynomial Functions

Polynomial functions are expressions that consist of variables raised to whole number powers, multiplied by coefficients. They are among the most basic and vital types of functions in mathematics. A polynomial function can be written in the form:

• \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \)

where \( a_n, a_{n-1},..., a_1, a_0 \) are constants, and \( n \) is a non-negative integer called the degree of the polynomial.
In the context of derivatives, each term in a polynomial can be differentiated using the power rule. For instance, consider \( y = 2x^3 \). It's a simple polynomial with just one term where \( a = 2 \) and \( n = 3 \). By applying the power rule, the derivative is \( 6x^2 \).
This is what makes polynomial functions easy to work with when performing differentiation – their structure allows for straightforward application of differentiation rules.

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative represents the rate at which the function's value changes as the input changes. It is like finding how fast something is moving or changing at any given point.
Differentiation has numerous applications in real-world scenarios such as physics, engineering, and economics. For the problem at hand, differentiation was used to find how the function \( y = 2x^3 \) changes with respect to \( x \). Calculating \( \frac{dy}{dx} = 6x^2 \) tells us the slope of the tangent line to the curve at any given point \( x \).
To perform differentiation, one commonly used rule is the power rule, essential for finding derivatives of polynomial functions. Understanding differentiation not only helps in solving mathematical problems but also gives insights into the behavior and trends of various phenomena.