Problem 11 - Calculus Early Transcendentals

Question

Find \(f^{\prime}(x)\) $$ f(x)=\frac{4}{\left(3 x^{2}-2 x+1\right)^{3}} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = \frac{-12(6x - 2)}{(3x^2 - 2x + 1)^4} \).

Step 1: Identify the Type of Function

The function given is a rational function of the form \( f(x) = \frac{g(x)}{h(x)} \), where \( g(x) = 4 \) is a constant and \( h(x) = \left(3x^2 - 2x + 1\right)^3 \). We will use the chain rule and the quotient rule to differentiate the function.

Step 2: Apply the Quotient Rule

The quotient rule states that for any functions \( u(x) \) and \( v(x) \), the derivative is \( \frac{d}{dx}\left( \frac{u}{v} \right) = \frac{v\cdot u' - u\cdot v'}{v^2} \). In our case, let \( u(x) = 4 \) and \( v(x) = (3x^2 - 2x + 1)^3 \). Then, \( u'(x) = 0 \) since \( u \) is a constant.

Step 3: Compute the Derivative of the Denominator

Determine \( v'(x) \) using the chain rule. First, let \( h(x) = 3x^2 - 2x + 1 \) and \( v(x) = h(x)^3 \). The derivative of \( v(x) \) is \( 3h(x)^2 \cdot h'(x) \). Calculate \( h'(x) = 6x - 2 \), leading to \( v'(x) = 3(3x^2 - 2x + 1)^2 \cdot (6x - 2) \).

Step 4: Compute the Derivative of the Function

Substitute the derivatives into the quotient rule formula: \( f'(x) = \frac{(3x^2 - 2x + 1)^3 \cdot 0 - 4 \cdot (3(3x^2 - 2x + 1)^2 \cdot (6x - 2))}{(3x^2 - 2x + 1)^6} \). Simplify this to get \( f'(x) = \frac{-12(3x^2 - 2x + 1)^2(6x - 2)}{(3x^2 - 2x + 1)^6} \).

Step 5: Simplify the Expression

Simplify the derivative expression by canceling terms. The final form is \( f'(x) = \frac{-12(6x - 2)}{(3x^2 - 2x + 1)^4} \).

Key Concepts

Chain Rule

The chain rule is a fundamental technique in calculus for finding the derivative of composite functions. If you have a function within another function, the chain rule is your go-to tool. It's expressed as follows:\[ (f \circ g)'(x) = f'(g(x)) \cdot g'(x) \]To apply the chain rule:

• Identify the "outer" function and the "inner" function. The outer function is the one being applied last, whereas the inner function is being operated on.
• Differentiate the outer function with respect to the inner function (as if the inner function was a simple variable).
• Multiply the result by the derivative of the inner function.

This technique is essential when calculating the derivative of complicated expressions such as powers, exponentials, and trigonometric functions that are nested together. In our exercise, we used the chain rule to differentiate \( v(x) = (3x^2 - 2x + 1)^3 \) by identifying the inner function \( h(x) = 3x^2 - 2x + 1 \) and the outer function being \( (h(x))^3 \). This method lets us break down complex operations into manageable parts.

Quotient Rule

The quotient rule is a handy formula used to differentiate functions that are fractions, where one function is divided by another. Given two functions, \( u(x) \) and \( v(x) \), the derivative of their quotient \( \frac{u}{v} \) is calculated using:\[ \frac{d}{dx}\left( \frac{u}{v} \right) = \frac{v \cdot u' - u \cdot v'}{v^2} \]Here's how to apply the quotient rule effectively:

• First, determine \( u(x) \) and \( v(x) \) from your function. \( u(x) \) is the numerator and \( v(x) \) is the denominator.
• Find the derivative of the numerator, \( u'(x) \), and the derivative of the denominator, \( v'(x) \).
• Substitute these values into the quotient rule formula: multiply \( v(x) \) by \( u'(x) \), and \( u(x) \) by \( v'(x) \).
• Subtract the second product from the first and divide the entire expression by \( v(x)^2 \).

In our example, the numerator is a simple constant, which made \( u'(x) = 0 \), simplifying the calculation. This illustrates how the quotient rule can simplify even seemingly complex function derivatives.

Rational Function Differentiation

Rational functions, which are ratios of two polynomials, are frequently encountered in calculus. Differentiating them often involves using rules like the chain rule and the quotient rule. Here's how you tackle such a problem:

• Identify the numerator and the denominator of the function.
• Apply the quotient rule since you have one polynomial over another.
• If the denominator is a composite function, as in our example, use the chain rule for its derivative.
• Simplify the expression after applying these rules to get the final derivative form.

In the given exercise, we dealt with a rational function: \( f(x) = \frac{4}{(3x^2 - 2x + 1)^3} \). We recognized that the core of calculating its derivative lies in properly applying the quotient rule and then using the chain rule to handle the complexity of the denominator. This systematic approach ensures that the differentiation of even complex rational functions can be tackled with confidence, leading to the correct and simplified derivative form.