Problem 18 - Calculus of a Single Variable

Question

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ y=2 x^{3}+6 x^{2}-1 $$

Step-by-Step Solution

Answer

The derivative of the function \( y = 2x^3 + 6x^2 - 1 \) is \( y' = 6x^2 + 12x \).

Step 1: Identify the terms and their powers

In the equation \( y = 2x^3 + 6x^2 - 1 \), there are three terms. The terms are \( 2x^3 \), \( 6x^2 \) and \( -1 \) with the powers 3, 2 and 0 respectively.

Step 2: Apply the power rule

Now, apply the power rule on each term. The power rule states if \( y = x^n \), then the derivative \( y' = n*x^{n-1} \). Hence, the derivative of \( 2x^3 \) is \( 3*2*x^{3-1} = 6x^2 \), the derivative of \( 6x^2 \) is \( 2*6*x^{2-1} = 12x \), and the derivative of constant \( -1 \) is zero, since the derivative of any constant is zero.

Step 3: Write down the derivative of the function

Combine the derivatives of the individual terms to obtain the derivative of the entire function. Hence, the derivative of the function \( y = 2x^3 + 6x^2 - 1 \) is given by \( y' = 6x^2 + 12x \)

Key Concepts

Differentiation Rules

Differentiation is a cornerstone concept in calculus, enabling us to find the rate at which a function is changing at any given point. To do this, we use certain rules known as differentiation rules. These rules were developed to simplify the process of finding derivatives, allowing us to systematically handle various types of functions without starting from scratch each time.
**Key Differentiation Rules:**

• **Constant Rule:** The derivative of a constant is zero. This is because a constant value does not change, so its rate of change is zero.
• **Sum Rule:** If you have a function that is a sum of two functions, the derivative of the sum is the sum of their derivatives. Mathematically, if \( f(x) = g(x) + h(x) \), then \( f'(x) = g'(x) + h'(x) \).
• **Power Rule:** This is crucial for polynomial functions and will be covered in detail in the next section.

By using these rules, you can break down complex functions into simpler parts and find their derivatives quickly.

Power Rule

The power rule is a fundamental technique in calculus for differentiating functions of the form \( x^n \). It's particularly powerful because it simplifies the differentiation process for polynomial terms. According to the power rule, if you have a term \( x^n \), its derivative is \( nx^{n-1} \). This means you multiply by the power and then reduce the power by one.
Let's break it down:

• If \( y = x^3 \), then by the power rule, \( y' = 3x^{3-1} = 3x^2 \).
• For a term like \( 2x^3 \), you multiply the coefficient by the power: \( 3*2 = 6 \), so the derivative is \( 6x^2 \).

This rule is incredibly useful when you are differentiating polynomials, as it allows you to handle each term independently and then combine the results.

Polynomial Functions

Polynomial functions are expressions that consist of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. They are foundational in many areas of mathematics because they are relatively simple and incredibly versatile.
**Key Characteristics of Polynomial Functions:**

• **Degree:** The highest power of the variable in the polynomial determines its degree. It helps in understanding the behavior and shape of the graph of the polynomial.
• **Form:** A basic polynomial takes the form \( a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \), where \( a_n, a_{n-1}, ..., a_0 \) are constants.
• **Differentiation:** Each term in a polynomial can be differentiated using the power rule, which makes finding the derivative straightforward. Once differentiated, the derivative of the polynomial is also a polynomial, but of one lesser degree.

This makes working with polynomials in calculus relatively easy, as their structure leads to predictable and manageable patterns.