Problem 12 - Calculus - AP Edition

Question

$$\text { Find } f^{\prime}(x)$$. $$f(x)=\frac{x-2}{x^{4}+x+1}$$

Step-by-Step Solution

Answer

\( f'(x) = \frac{-3x^4 + 8x^3 - 2}{(x^4 + x + 1)^2} \)

Step 1: Identify the Function Type

The given function \( f(x) = \frac{x-2}{x^4 + x + 1} \) is a rational function. This means it is expressed as a fraction, with a polynomial in the numerator and another polynomial in the denominator.

Step 2: Recall the Quotient Rule

The quotient rule is used to differentiate functions of the form \( \frac{u}{v} \), where \( u \) and \( v \) are both differentiable functions of \( x \). The quotient rule is given by: \[ f'(x) = \frac{u'v - uv'}{v^2}. \] Here, \( u = x - 2 \) and \( v = x^4 + x + 1 \).

Step 3: Differentiate the Numerator

Find the derivative of the numerator \( u = x - 2 \). The derivative is: \( u' = 1 \), since the derivative of \( x \) is 1 and the derivative of a constant is 0.

Step 4: Differentiate the Denominator

Find the derivative of the denominator \( v = x^4 + x + 1 \). The derivative is: \( v' = 4x^3 + 1 \), since the derivative of \( x^4 \) is \( 4x^3 \), the derivative of \( x \) is 1, and the derivative of the constant is 0.

Step 5: Apply the Quotient Rule

Substitute \( u' \), \( v' \), \( u \), and \( v \) into the quotient rule formula. Thus: \[f'(x) = \frac{(1)(x^4 + x + 1) - (x - 2)(4x^3 + 1)}{(x^4 + x + 1)^2}. \]

Step 6: Simplify the Expression

Simplify the expression obtained from the quotient rule. Distribute the terms in the numerator to get: \[ f'(x) = \frac{x^4 + x + 1 - (4x^4 - 8x^3 + x - 2)}{(x^4 + x + 1)^2}. \] Further simplifying gives: \[ f'(x) = \frac{-3x^4 + 8x^3 - 2}{(x^4 + x + 1)^2}. \]

Key Concepts

Quotient Rule

The quotient rule is a fundamental concept in calculus used to find the derivative of a function that is the quotient of two differentiable functions. If you have a function expressed as \( f(x) = \frac{u(x)}{v(x)} \), where both \( u(x) \) and \( v(x) \) are differentiable, the quotient rule helps! The formula is:

• \( f'(x) = \frac{u'v - uv'}{v^2} \)

You start by identifying which parts of your function correspond to \( u(x) \) and \( v(x) \). Then, differentiate these parts separately: \( u'(x) \) and \( v'(x) \). Substitute these into the quotient rule formula. Here, we applied the quotient rule to find the derivative of \( f(x) = \frac{x-2}{x^4+x+1} \) by setting \( u = x - 2 \) and \( v = x^4 + x + 1 \). The quotient rule effectively allows the separation of the complex derivative process into more manageable parts. When you practice using the quotient rule, you’ll find that it's a powerful tool for tackling derivatives involving rational functions.

Differentiation Techniques

When tackling calculus problems, knowing various differentiation techniques is essential. For rational functions, differentiating involves breaking the problem into smaller parts using rules like the power rule and quotient rule.

• Power Rule: This rule states that if you have \( x^n \), its derivative \( nx^{n-1} \). This is used for differentiating polynomials. For example, in \( x^4 \), the derivative would be \( 4x^3 \).
• Constant Rule: The derivative of any constant is zero. It's helpful in simplifying steps.
• Combining Rules: In rational functions, both the quotient rule and power rule might be needed, as seen in \( f(x) = \frac{x-2}{x^4+x+1} \). Differentiating involves calculating \( u' \) and \( v' \), where using these rules can make your task easier.

Understanding and applying these techniques lets you navigate through the differentiation of complex functions much more smoothly. Each rule is a building block towards mastering calculus.

Rational Function

A rational function is a ratio of two polynomials. It takes the form \( f(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \).

• Numerator and Denominator: In the given function \( f(x) = \frac{x-2}{x^4 + x + 1} \), \( x-2 \) is the numerator, while \( x^4 + x + 1 \) is the denominator.
• Properties: Rational functions can be continuous and differentiable where their denominator is nonzero. They often exhibit vertical or horizontal asymptotes, influencing their graph's shape.
• Derivatives: Finding derivatives of rational functions often involves the quotient rule, as these functions are typically expressed as one polynomial divided by another. The quotient rule simplifies the differentiation process by providing a structured formula involving the derivatives of both \( P(x) \) and \( Q(x) \).

Grasping rational functions and understanding their components is critical for solving related calculus problems, as it forms the basis for applying the quotient rule and other differentiation strategies.