Problem 25 - Calculus an Applied Approach

Question

Use the General Power Rule to find the derivative of the function. $$ g(x)=(4-2 x)^{3} $$

Step-by-Step Solution

Answer

The derivative of the function \(g(x) = (4 - 2x)^3\) is \(g'(x) = -6 (4 - 2x)^2\).

Step 1: Identify the function and its parts

First, recognize the function as \(g(x) = (4 - 2x)^3\). Here, \(u = 4 - 2x\) and \(n = 3\). The application of the General Power Rule requires identifying these components.

Step 2: Calculate the derivative of u

The derivative of \(u = 4 - 2x\) with respect to x is \(u' = -2\). This is obtained using the basic power rule.

Step 3: Apply the General Power Rule

Apply the General Power Rule, which is \((u^n)' = n u^{n-1} \cdot u'\). Substituting n = 3, \(u = 4 - 2x\), and \(u' = -2\) we get: \[ g'(x) = 3 (4 - 2x)^{2} \cdot (-2) \].

Step 4: Simplify the evaluation

Now, simply the expression to its smallest form: \[ g'(x) = -6 (4 - 2x)^{2} \].

Key Concepts

General Power Rule

The General Power Rule is a helpful tool in calculus for finding derivatives of functions raised to a power. It's a slight extension of the basic Power Rule, tailored to manage functions that aren't just simple variables.
The formula for the General Power Rule is expressed as \((u^n)' = n \cdot u^{n-1} \cdot u'\), where:

• \(u\) is a function of \(x\)
• \(n\) is a constant, representing the power that the function \(u\) is raised to
• \(u'\) is the derivative of \(u\) with respect to \(x\)

In our given function \(g(x) = (4 - 2x)^3\), we recognize \(u\) as \(4 - 2x\) with \(n = 3\). First, you calculate the derivative of \(u\) which comes out to be \(u' = -2\). After these components are identified, you can use this rule to find the derivative efficiently. The beauty of this rule is that it allows you to differentiate complex expressions cleanly by handling one part at a time. Apply the derivatives and simplify to get an answer without overwhelming calculations.

Derivative

A derivative in calculus is a way to show how a function changes as its input changes. Simply put, it tells us the rate at which a function is changing at any given point.
The derivative provides critical information about the behavior of functions. For instance, it helps in finding:

• The slope of a function at a particular point
• Maximum and minimum values of functions
• Points of inflection where a function changes the direction of its curvature

To better understand, consider the function \(g(x) = (4 - 2x)^3\). Finding its derivative entails using rules of differentiation to express how \(g\) changes concerning \(x\). This process involves simplifying expressions, which often includes employing algebraic techniques and specific derivative rules, such as the General Power Rule in our example. By understanding derivatives, you can interpret changes within various scenarios and solve complex real-world problems.

Power Rule

The Power Rule is a fundamental rule in calculus that simplifies the process of differentiation for polynomial functions. It's straightforward and can be a handy tool for beginners trying their hand at differentiating simple power expressions.
The basic Power Rule formula is \(\frac{d}{dx} x^n = nx^{n-1}\), where:

• \(x\) is the variable
• \(n\) is a real number constant, denoting the power

This rule states that when you're finding the derivative of \(x^n\), you multiply \(n\) by the expression \(x\) raised to the power of \(n-1\). It's simple yet powerful, allowing many functions to be differentiated quickly.
In the context of our exercise, although the function isn't a simple polynomial like \(x^n\), the Power Rule concept is instrumental in finding the derivative of \(u = 4 - 2x\). We used it to compute \(u' = -2\), which simplifies our task when applying the General Power Rule.
Understanding these rules not only helps in solving textbook exercises but also paves the way to tackle more advanced calculus topics.