Problem 36 - Brief Applied Calculus

Question

Use the Generalized Power Rule to find the derivative of each function. $$ f(x)=2 x\left(x^{3}-1\right)^{4} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 2(x^3 - 1)^4 + 24x^3(x^3 - 1)^3 \).

Step 1: Understand the Function

We are given the function \( f(x) = 2x(x^3 - 1)^4 \). It consists of two parts, a monomial \( 2x \) and a composite function \( (x^3 - 1)^4 \). The method we'll use here is the Product Rule, teamed up with the Generalized Power Rule because the function is a product of two sub-functions.

Step 2: Apply the Product Rule

According to the Product Rule for differentiation, if \( u(x) \) and \( v(x) \) are functions, then the derivative of their product \( u(x)v(x) \) is \( u'(x)v(x) + u(x)v'(x) \). Here, let \( u(x) = 2x \) and \( v(x) = (x^3 - 1)^4 \). This means we need to find \( u'(x) \) and \( v'(x) \).

Step 3: Differentiate \( u(x) = 2x \)

The derivative of \( u(x) = 2x \) is \( u'(x) = 2 \). This is straightforward since it is a simple linear function.

Step 4: Differentiate \( v(x) = (x^3 - 1)^4 \) using the Generalized Power Rule

Using the Generalized Power Rule: for a function of the form \( (g(x))^n \), the derivative is \( n(g(x))^{n-1}g'(x) \). Here, \( g(x) = x^3 - 1 \) and \( n = 4 \). Thus, \( g'(x) = 3x^2 \). Therefore, \( v'(x) = 4(x^3 - 1)^3 \cdot 3x^2 = 12x^2(x^3 - 1)^3 \).

Step 5: Substitute into the Product Rule

Substitute \( u(x) \), \( u'(x) \), \( v(x) \), and \( v'(x) \) back into the Product Rule formula: \( f'(x) = u'(x)v(x) + u(x)v'(x) \). This gives: \( f'(x) = 2(x^3 - 1)^4 + 2x(12x^2(x^3 - 1)^3) \).

Step 6: Simplify the Expression

Simplify: \( f'(x) = 2(x^3 - 1)^4 + 24x^3(x^3 - 1)^3 \). This derivative can be left as is unless further factorization or simplification is required by context.

Key Concepts

Product Rule

The Product Rule is a fundamental method used in calculus to differentiate functions that are multiplied together. Imagine you have two functions, let’s call them \( u(x) \) and \( v(x) \). The Product Rule states that the derivative of their product \( u(x)v(x) \) is given by:\[ f'(x) = u'(x)v(x) + u(x)v'(x) \]This means you take the derivative of the first function \( u(x) \), multiply it by the second function \( v(x) \), then add the product of the first function \( u(x) \) and the derivative of the second function \( v(x) \).When you apply the Product Rule as shown in the exercise, you can see it in action with \( u(x) = 2x \) and \( v(x) = (x^3 - 1)^4 \). By meticulously calculating each derivative, you can successfully differentiate the entire function. This rule is essential as it allows the differentiation of complex products easily.

Generalized Power Rule

The Generalized Power Rule extends the basic power rule for differentiation to composite functions. Typically, the power rule says that the derivative of \( x^n \) is \( nx^{n-1} \). The generalized form applies when you have a function \( (g(x))^n \), where \( g(x) \) is a differentiable function.Using this rule, the derivative \[ \frac{d}{dx}(g(x))^n = n(g(x))^{n-1}g'(x) \]is calculated by first applying the power \( n \), then reducing the power by one, and finally multiplying by the derivative of the inside function \( g(x) \).In the original exercise, the function \( v(x) = (x^3 - 1)^4 \) was differentiated by setting \( g(x) = x^3 - 1 \) and \( n = 4 \). By computing \( g'(x) = 3x^2 \), and multiplying this by the adjusted power expression \( 4(x^3 - 1)^3 \), you get the derivative \( v'(x) \). This method simplifies the process for differentiating nested functions.

Differentiation Techniques

Differentiation techniques are tools that make the process of finding derivatives manageable and efficient. Among these techniques, the Product Rule and the Generalized Power Rule play crucial roles. Differentiation involves:

• Recognizing the type of function or combination of functions you are dealing with.
• Choosing the best differentiation rule or technique that applies.
• Executing the rule carefully to ensure accuracy.

Other techniques include the Chain Rule, Quotient Rule, and basic rules for polynomials and exponentials. Each of these techniques is applied based on the pattern and arrangement of terms in the function. By combining these methods, particularly the Product Rule and the Generalized Power Rule, you get a structured approach for tackling complex differentiation problems. They allow for breaking down complicated expressions into simpler parts, evaluating their derivatives, and building them back into a coherent final answer.