Problem 117 - Calculus Volume 1

Question

Find \(f^{\prime}(x)\) for each function. $$ f(x)=\frac{x+9}{x^{2}-7 x+1} $$

Step-by-Step Solution

Answer

\( f^{\prime}(x) = \frac{-x^2 - 18x + 64}{(x^2 - 7x + 1)^2} \)

Step 1: Identify the Differentiation Rule Needed

The function given, \( f(x) = \frac{x+9}{x^2-7x+1} \), is a rational function because it is the quotient of two polynomials. Therefore, we need to apply the quotient rule for differentiation.

Step 2: Recall the Quotient Rule Formula

The quotient rule states that if \( f(x) = \frac{g(x)}{h(x)} \), then the derivative, \( f^{\prime}(x) \), is given by \[ f^{\prime}(x) = \frac{g^{\prime}(x) h(x) - g(x) h^{\prime}(x)}{[h(x)]^2} \]. We will apply this rule to find \( f^{\prime}(x) \).

Step 3: Differentiate the Numerator and Denominator

Identify the numerator as \( g(x) = x + 9 \) and the denominator as \( h(x) = x^2 - 7x + 1 \). Compute their derivatives:- The derivative of the numerator \( g^{\prime}(x) = 1 \), since the derivative of \( x \) is 1 and 9 is a constant.- The derivative of the denominator \( h^{\prime}(x) = 2x - 7 \), using the power rule.

Step 4: Substitute into the Quotient Rule Formula

Substitute into the quotient rule:\[ f^{\prime}(x) = \frac{(1)(x^2-7x+1) - (x+9)(2x-7)}{(x^2-7x+1)^2} \].

Step 5: Simplify the Expression

First translate the expression into individual products and collect like terms:- The first term is \( 1 \cdot (x^2 - 7x + 1) = x^2 - 7x + 1 \).- The second term, using distribution, is \( (x + 9)(2x - 7) = 2x^2 - 7x + 18x - 63 = 2x^2 + 11x - 63 \).The resulting expression is\( f^{\prime}(x) = \frac{x^2 - 7x + 1 - (2x^2 + 11x - 63)}{(x^2-7x+1)^2} \).

Step 6: Combine and Simplify Like Terms

Simplify the numerator by combining like terms:- Combine the \( x^2 \) terms: \( x^2 - 2x^2 = -x^2 \).- Combine the linear terms: \( -7x - 11x = -18x \).- Combine the constant terms: \( 1 + 63 = 64 \).Thus, the derivative is\[ f^{\prime}(x) = \frac{-x^2 - 18x + 64}{(x^2-7x+1)^2} \].

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus, essential for understanding how functions behave. It involves finding the rate at which a function changes at any given point and is denoted by the derivative. For any real-valued function, the derivative can show the slope of the tangent line to the graph at a specific point, providing insight into the function's growth or decline.

• It gives a mathematical explanation of change.
• It's applied in various fields, including physics, engineering, and economics.

In the context of finding the derivative of a function like the one given in the original exercise, the differentiation process involves identifying how both the numerator and the denominator contribute to the rate of change. Here, we apply the quotient rule, specifically designed for differentiating rational functions consisting of quotients of two simpler functions known as polynomials.

Rational Functions

Rational functions are expressions that result from the division of two polynomials. For the function provided in the exercise, the rational function is formed by the numerator, a linear polynomial, and the denominator, a quadratic polynomial. This structure gives the function specific characteristics and makes it a prime candidate for using the quotient rule in differentiation.

• The numerator of our function is a simple linear polynomial, easy to differentiate.
• The denominator is a quadratic polynomial, common in rational functions, and requires using the power rule for its derivative.

Rational functions can exhibit asymptotic behavior as they approach undefined values due to the denominator. When differentiating these functions, it's important to apply the rules correctly to capture these subtleties in behavior.

Polynomial Derivatives

Polynomial derivatives are straightforward to compute since they follow easily memorized rules. For each term in a polynomial:

• Use the power rule: differentiate by bringing down the exponent as a coefficient and subtracting one from the exponent.

For example, consider the polynomial used as the denominator in the exercise: \( h(x) = x^2 - 7x + 1 \):

• The derivative of \( x^2 \) is \( 2x \).
• The derivative of \( -7x \) is \( -7 \).
• Constants, like \( +1 \), have a derivative of \( 0 \).

Hence, the complete derivative is \( h'(x) = 2x - 7 \). Understanding these derivatives is critical for successfully applying the quotient rule and finding the rate of change in more complex rational functions.