Problem 12 - Calculus Single Variable

Question

Calculate \(g^{\prime}(x)\) by using the formulas and rules that are summarized at the end of this section. $$ g(x)=x^{2}-4 x $$

Step-by-Step Solution

Answer

The derivative is \( g'(x) = 2x - 4 \).

Step 1: Identify the Function Components

The function given is \( g(x) = x^2 - 4x \). This function is a polynomial consisting of two terms: \( x^2 \) and \( -4x \).

Step 2: Apply the Power Rule

To differentiate \( x^2 \), apply the power rule: \( \frac{d}{dx}[x^n] = n \cdot x^{n-1} \). Here, \( n = 2 \), so \( \frac{d}{dx}[x^2] = 2x^{2-1} = 2x \).

Step 3: Apply the Constant Multiplier Rule

Differentiate \( -4x \) using the constant multiplier rule: \( \frac{d}{dx}[cx] = c \). The derivative of \( -4x \) is \( -4 \) since the constant multiplier \( c \) is \( -4 \).

Step 4: Combine the Derivatives

Add the derivatives of each term to find \( g'(x) \). So, \( g'(x) = 2x - 4 \).

Key Concepts

Power Rule

The power rule is a fundamental principle in calculus for finding the derivative of power functions. It simplifies the differentiation process for terms of the form \( x^n \). To apply the power rule, follow these simple steps:

• Identify the exponent \( n \) in the term \( x^n \).
• Multiply the entire term by the exponent \( n \).
• Reduce the exponent by 1 to get the new exponent.

For example, in the term \( x^2 \), \( n \) is 2. Therefore, according to the power rule, its derivative is calculated as follows:
- Multiply 2 by \( x^{2-1} \), resulting in \( 2x \).
This straightforward approach makes the power rule particularly useful for polynomial differentiation, as we'll see in the next section.

Polynomial Differentiation

In calculus, polynomial differentiation involves finding the derivative of a polynomial function. Polynomial functions are expressions involving one or multiple terms of powers of \( x \), such as \( g(x) = x^2 - 4x \) in the original exercise.
To differentiate a polynomial function:

• Split the function into its individual terms. For example, separate \( x^2 \) and \( -4x \).
• Apply differentiation rules, such as the power rule or constant multiplier rule, to each term individually.
• Add the results for each term to find the overall derivative of the polynomial.

Using these steps ensures you can straightforwardly and efficiently find the derivative, especially when dealing with functions that contain multiple terms.

Constant Multiplier Rule

The constant multiplier rule is a handy tool when differentiating terms involving constant coefficients. It simplifies the differentiation of terms where a constant is multiplied by a function. Follow these steps to apply the constant multiplier rule:

• Identify the constant in front of a term. For example, in \( -4x \), the constant is -4.
• Take the derivative of the variable part (for example, \( x \) becomes 1).
• Multiply the constant by the derivative of the variable part.

Let's differentiate \( -4x \), where the constant is -4 and the variable is \( x \).
- Derivative of \( x \) is 1.- Hence, multiply 1 by -4, resulting in -4.
After applying this rule, you add it to other differentiated terms to form a complete picture of the function's derivative. This method ensures calculations remain simple, even when dealing with complex polynomials.