Problem 26 - Brief Applied Calculus

Question

Use the Generalized Power Rule to find the derivative of each function. $$ f(x)=\left(x^{2}+4\right)^{3}-\left(x^{2}+4\right)^{2} $$

Step-by-Step Solution

Answer

The derivative is \( 2x(x^2 + 4)(3x^2 + 10) \).

Step 1: Understand the Function

The function given is \( f(x) = (x^2 + 4)^3 - (x^2 + 4)^2 \). This is a combination of two expressions which both involve powers of the same inner function, \( x^2 + 4 \).

Step 2: Apply the Generalized Power Rule

The Generalized Power Rule states that the derivative of \( [g(x)]^n \) with respect to \( x \) is \( n[g(x)]^{n-1}g'(x) \). We'll apply this rule to each part separately: \( (x^2 + 4)^3 \) and \( (x^2 + 4)^2 \).

Step 3: Differentiate the First Term

For \( (x^2 + 4)^3 \):- \( n = 3 \), \( g(x) = x^2 + 4 \), and \( g'(x) = 2x \).- Using the rule, the derivative is \( 3(x^2 + 4)^{2} \cdot 2x = 6x(x^2 + 4)^2 \).

Step 4: Differentiate the Second Term

For \( (x^2 + 4)^2 \):- \( n = 2 \), \( g(x) = x^2 + 4 \), and \( g'(x) = 2x \).- Using the rule, the derivative is \( 2(x^2 + 4)^{1} \cdot 2x = 4x(x^2 + 4) \).

Step 5: Subtract the Derivatives

The derivative of the entire function is the derivative of the first term minus the derivative of the second term:\[ f'(x) = 6x(x^2 + 4)^2 - 4x(x^2 + 4) \].

Step 6: Simplify the Expression

Factor out \( 2x(x^2 + 4) \) from both terms:\[ f'(x) = 2x(x^2 + 4)[3(x^2 + 4) - 2] = 2x(x^2 + 4)(3x^2 + 12 - 2) \].Simplify inside the brackets:\[ f'(x) = 2x(x^2 + 4)(3x^2 + 10) \].

Key Concepts

Generalized Power Rule

To understand the Generalized Power Rule, we first recognize that it is an extension of the basic power rule in calculus, used to differentiate functions of the form \((g(x))^n\). The rule states that the derivative of such a function is given by:\[n(g(x))^{n-1}g'(x)\]where:

• \(n\) is the power to which the function \(g(x)\) is raised.
• \(g(x)\) is the inner function.
• \(g'(x)\) is the derivative of the inner function \(g(x)\).

In the original exercise, this rule helps us break down the differentiation problem into simpler parts. Applying it to each component, like \((x^2 + 4)^3\) and \((x^2 + 4)^2\), starts by identifying these as power functions with respect to the inner function \(x^2 + 4\).
The steps involve determining \(g(x)\) and computing its derivative, followed by plugging these values into the generalized formula. This approach greatly simplifies the process when dealing with more complex composite functions.

Function Differentiation

Function differentiation involves finding the derivative, which represents the rate at which a function changes at any given point. In our case, we are differentiating a function made of two terms: \((x^2 + 4)^3\) and \((x^2 + 4)^2\). Each of these terms consists of the same inner function, \(x^2 + 4\), making it an ideal candidate for the application of the Generalized Power Rule.
For differentiation:

• The first term, \((x^2 + 4)^3\), is differentiated as follows: Identify \(n = 3\), \(g(x) = x^2 + 4\), and compute \(g'(x) = 2x\).
• Apply the rule to find the derivative: \(3(x^2 + 4)^2 \, \cdot \, 2x = 6x(x^2 + 4)^2\).
• The second term, \((x^2 + 4)^2\), is differentiated similarly: Here, \(n = 2\), \(g(x) = x^2 + 4\), and again, \(g'(x) = 2x\).
• Its derivative becomes: \(2(x^2 + 4) \, \cdot \, 2x = 4x(x^2 + 4)\).

This systematic approach allows for accurate calculation of the derivative of complex expressions by breaking them into manageable pieces.

Simplification of Derivatives

After applying the Generalized Power Rule to differentiate each term, the next step involves simplifying the expression for the derivative of the entire function. In our example, we find:

• The derivative of the first term: \(6x(x^2 + 4)^2\).
• The derivative of the second term: \(4x(x^2 + 4)\).

Subtracting these derivatives gives us:\[f'(x) = 6x(x^2 + 4)^2 - 4x(x^2 + 4)\]
To simplify, we factor out common terms. Here, \(2x(x^2 + 4)\) is common, so factoring it out leads to:\[f'(x) = 2x(x^2 + 4)[3(x^2 + 4) - 2]\]
After simplifying inside the brackets, we arrive at the complete simplified derivative:\[f'(x) = 2x(x^2 + 4)(3x^2 + 10)\]
This simplification process not only streamlines the function but also ensures the accuracy and clarity of the mathematical expression.