Problem 38 - Brief Applied Calculus

Question

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

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 2(2x-1)^2(2x+1)^3(14x-1) \).

Step 1: Identify the Structure

The function given is a product of two functions raised to a power: \((2x-1)^3\) and \((2x+1)^4\). The derivative of a product of functions can be found using the Product Rule: If \(u(x)\) and \(v(x)\) are functions, then \((u \, v)\)' = u'v + uv'. Here, let \( u(x) = (2x-1)^3 \) and \( v(x) = (2x+1)^4 \).

Step 2: Differentiate Each Function

First, apply the Generalized Power Rule to \( u(x) = (2x-1)^3 \). The derivative is: \( u'(x) = 3(2x-1)^2 \cdot (2) = 6(2x-1)^2 \). Now, apply the Generalized Power Rule to \( v(x) = (2x+1)^4 \). The derivative is: \( v'(x) = 4(2x+1)^3 \cdot (2) = 8(2x+1)^3 \).

Step 3: Apply the Product Rule

Using the Product Rule, the derivative \( f'(x) \) is: \( f'(x) = u'(x)v(x) + u(x)v'(x) \). Substitute \( u(x), u'(x), v(x), \) and \( v'(x) \). \( f'(x) = [6(2x-1)^2] \cdot [(2x+1)^4] + [(2x-1)^3] \cdot [8(2x+1)^3] \).

Step 4: Simplify the Expression

Combine and simplify the expression: \( f'(x) = 6(2x-1)^2 (2x+1)^4 + 8(2x-1)^3 (2x+1)^3 \). Both terms contain factors of \((2x-1)^2\) and \((2x+1)^3\), so factor these out. Then simplify: \( f'(x) = 2(2x-1)^2(2x+1)^3 [3(2x+1) + 4(2x-1)] \).Calculate the remaining bracketed expression: \( 3(2x+1) + 4(2x-1) = 6x + 3 + 8x - 4 = 14x - 1 \).Thus, \( f'(x) = 2(2x-1)^2(2x+1)^3(14x-1) \).

Key Concepts

Generalized Power Rule

The Generalized Power Rule is a powerful tool used in calculus for differentiating functions that are raised to a power. To apply it, you take the exponent as a coefficient, reduce the original power by one, and then multiply by the derivative of the inside function. This helps simplify the process of finding derivatives for more complex functions.

For example, consider the function given in the exercise, which is \[ f(x) = (2x - 1)^3 (2x + 1)^4 \]. This function consists of two parts, each raised to its unique power.

• The derivative of \((2x-1)^3\) involves multiplying the power 3 by the derivative of \(2x-1\), which is 2. This results in \(6(2x - 1)^2\).
• Similarly, for \((2x+1)^4\), you multiply the power 4 by the derivative of \(2x+1\), which is also 2. Thus, you get \(8(2x + 1)^3\).

This simplification is incredibly useful when dealing with functions that might otherwise be overwhelming if handled directly without rules like the Generalized Power Rule.

Product Rule

When you encounter the product of two functions in calculus, to differentiate, you use the Product Rule. This rule states that if you have two functions, say \(u(x)\) and \(v(x)\), their derivative is given by:\[ (u \, v)' = u'v + uv' \]This means you differentiate the first function, multiply by the second, add it to the product of the second function’s derivative and the first function.

In the exercise, the function \(f(x) = (2x-1)^3 (2x+1)^4\) is a product of \(u(x) = (2x-1)^3\) and \(v(x) = (2x+1)^4\). To apply the Product Rule, you will:

• First differentiate \(u(x)\) to get \(u'(x) = 6(2x-1)^2\).
• Differentiate \(v(x)\) to get \(v'(x) = 8(2x+1)^3\).
• Plug these into the Product Rule formula: \(f'(x) = u'(x)v(x) + u(x)v'(x)\).

This results in two terms that when added together, give you the derivative of the entire function.

Derivative Simplification

After you have used the Generalized Power Rule and Product Rule to find the derivative, the next step is simplification. This step ensures the derivative is in its simplest and most understandable form, and often involves factoring and combining like terms.

In the final step of the solution, you simplify the expression resulting from the Product Rule:\[ f'(x) = 6(2x-1)^2 (2x+1)^4 + 8(2x-1)^3 (2x+1)^3 \]Both terms share common factors \((2x-1)^2\) and \((2x+1)^3\), which can be factored out.

• This simplification turns the expression into \(2(2x-1)^2(2x+1)^3 [3(2x+1) + 4(2x-1)]\).
• Further complete the simplification inside the bracket: \(3(2x+1) + 4(2x-1) = 14x - 1\).

Thus, the final simplified derivative is:\[ f'(x) = 2(2x-1)^2(2x+1)^3(14x-1) \]Through simplification, you obtain a cleaner, more manageable expression that reflects the function's behavior accurately.