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Chapter 3: Problem 44

Find the derivatives of the given functions. Assume that \(a, b, c,\) and \(k\) are constants. $$f(x)=\frac{a x+b}{x}$$

Short Answer

Expert verified
The derivative of \( f(x) = \frac{a x + b}{x} \) is \( f'(x) = -\frac{b}{x^2} \).

Step by step solution

01

Identify the Problem

We need to find the derivative of the function \( f(x) = \frac{a x + b}{x} \). This is a rational function, and we will use the quotient rule to find its derivative.
02

Rewrite the Function

Rewrite the function \( f(x) = \frac{a x + b}{x} \) as \( f(x) = a + \frac{b}{x} \) by separating the terms.
03

Differentiate Using Individual Rules

The function is \( f(x) = a + \frac{b}{x} \). Differentiate it term by term: - The derivative of the constant \( a \) is 0.- The derivative of \( \frac{b}{x} \), which is \( b \cdot x^{-1} \), is \( -b \cdot x^{-2} \), or equivalently \(-\frac{b}{x^2}\).
04

Combine the Results

Combine the derivatives of the separated terms: - The derivative \( 0 \) from the constant and - The derivative \( -\frac{b}{x^2} \) from the term \( \frac{b}{x} \).Hence, the derivative of \( f(x) \) is \( f'(x) = -\frac{b}{x^2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quotient Rule
In calculus, the Quotient Rule is a method used to find the derivative of a function that is the division of two other functions. The general formula for the quotient rule for two differentiable functions, say \(u(x)\) and \(v(x)\), is:\[\left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2}\]To apply it effectively:
  • The function in the numerator is \(u(x)\), and its derivative is \(u'(x)\).
  • The function in the denominator is \(v(x)\), and its derivative is \(v'(x)\).
This technique helps manage complex rational functions efficiently. For example, with \(f(x) = \frac{a x + b}{x}\), it initially seemed necessary to apply the quotient rule, but simplification revealed a combination of simpler terms that are differentiated more straightforwardly.
Rational Function
A rational function is any function which can be expressed as the ratio of two polynomials. It takes the form:\[f(x) = \frac{p(x)}{q(x)}\]where \(p(x)\) and \(q(x)\) are polynomials. The function \(f(x) = \frac{a x + b}{x}\) is an example. Here:
  • \(p(x) = a x + b\)
  • \(q(x) = x\)
Rational functions are defined wherever the denominator is not zero, making them versatile but occasionally tricky when dealing with zero points. Analyzing rational functions often involves using techniques like the quotient rule, especially when differentiating, but as seen in the original problem, simplification can lead to easier differentiation.
Differentiation
Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to a variable. When differentiating, two main rules are vital:
  • The Power Rule: If \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\).
  • The Sum Rule: If \(f(x) = g(x) + h(x)\), then \(f'(x) = g'(x) + h'(x)\).
In the exercise given, we derived the derivative by first re-writing the function into a simpler form: \(f(x) = a + \frac{b}{x}\), and then applying these basic differentiation rules to each term. This process enables us to find the derivative \(f'(x) = -\frac{b}{x^2}\). Additionally, understanding differentiation can assist in analyzing the behavior of functions more deeply, such as determining where a function is increasing or decreasing.
Constant Function
A constant function is a simple yet powerful concept in calculus. It is a function whose output value is the same, regardless of the input value, typically expressed as \(f(x) = c\) where \(c\) is a constant. A key differentiation rule is:
  • The derivative of a constant is always zero.
In the problem, the constant \(a\) was treated in this manner. When differentiating, the term \(a\) (as part of \(f(x) = a + \frac{b}{x}\)) results in a derivative of zero, reflecting no change over its domain. This simplicity is vital in breaking down more complex functions into manageable parts, allowing focus on non-constant components that vary with \(x\).

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Most popular questions from this chapter

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