Problem 117 - Calculus Volume 1

Question

For the following exercises, find \(f^{\prime}(x)\) for each function. $$f(x)=\frac{x+9}{x^{2}-7 x+1}$$

Step-by-Step Solution

Answer

\(f'(x) = \frac{-x^2 + 11x - 62}{(x^2 - 7x + 1)^2}\).

Step 1: Identify the Derivative Rule

The function is a quotient, so we will use the quotient rule for differentiation. For a function of the form \(\frac{u(x)}{v(x)}\), the derivative \(f'(x)\) is given by \((u'v - uv')/v^2\).

Step 2: Define the Functions

Let \(u(x) = x + 9\) and \(v(x) = x^2 - 7x + 1\). We need to find \(u'(x)\) and \(v'(x)\) as part of using the quotient rule.

Step 3: Differentiate the Numerator

Differentiate \(u(x) = x + 9\). The derivative \(u'(x)\) is simply the derivative of \(x\), so \(u'(x) = 1\).

Step 4: Differentiate the Denominator

Differentiate \(v(x) = x^2 - 7x + 1\). Using the power rule, \(v'(x)\) is \(2x - 7\).

Step 5: Apply the Quotient Rule

Substitute \(u(x)\), \(u'(x)\), \(v(x)\), and \(v'(x)\) into the quotient rule formula: \(f'(x) = \frac{(1)(x^2 - 7x + 1) - (x + 9)(2x - 7)}{(x^2 - 7x + 1)^2}\).

Step 6: Simplify the Expression

Simplify the expression:1. Expand: \((x^2 - 7x + 1) - (x+9)(2x-7) = x^2 - 7x + 1 - (2x^2 - 7x + 18x - 63)\).2. Combine like terms: \(x^2 - 7x + 1 - 2x^2 + 18x - 63\) simplifies to \(-x^2 + 11x - 62\).3. The derivative is \(f'(x) = \frac{-x^2 + 11x - 62}{(x^2 - 7x + 1)^2}\).

Key Concepts

Quotient Rule

When differentiating functions that involve division, such as a fraction of two functions, we use the **Quotient Rule.**
It's a powerful technique that allows us to find the derivative of a quotient of two functions.
The Quotient Rule formula is:

• If you have a function \(f(x) = \frac{u(x)}{v(x)}\), then its derivative \(f'(x)\) is given by \(\frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}\).

Think of it as a structured way to handle fractions when differentiating.
First, differentiate the numerator function \(u(x)\) to get \(u'(x)\).
Then, differentiate the denominator function \(v(x)\) to get \(v'(x)\).
Substitute these derivatives into the Quotient Rule formula.
By arranging these components as outlined by the Quotient Rule, you ensure correct differentiation even for complex ratios.

Differentiation Techniques

In calculus, **differentiation techniques** are methods used to calculate the rate at which a function is changing at any given point.
Different functions require different rules for differentiation. Knowing which technique to apply can make the process straightforward.
Key techniques include:

• Product Rule: Used for products of two functions.
• Quotient Rule: Used for quotients of two functions, as seen in this exercise.
• Chain Rule: Useful for compositions of functions.

Choosing the right technique depends on the structure of the function you are dealing with.
By systematically analyzing functions, you can determine the most efficient method to find the derivative.
This allows for precision and efficiency in calculus problem solving.

Power Rule

The **Power Rule** is one of the most commonly used differentiation techniques, particularly because of its simplicity and wide applicability.
It states that for a function of the form \(x^n\), where \(n\) is a real number, the derivative is \(nx^{n-1}\). This rule provides a quick way to differentiate polynomial expressions.
In our exercise, we encounter the Power Rule while differentiating \(v(x) = x^2 - 7x + 1\).

• The derivative of \(x^2\) is \(2x\),
• Using the Power Rule on \(-7x\) results in \(-7\).
• Constants like \(+1\) vanish, since their derivative is zero.

Being handy with the Power Rule allows for a swift computation of the derivatives of terms in polynomial functions.
It's essential for efficiently addressing many calculus problems and complements other differentiation techniques like the Quotient Rule.