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Chapter 7: Problem 5

Find \(f^{\prime}(x)\) if \(f(x)\) is the given expression. $$ \ln |3-2 x| $$

Short Answer

Expert verified
The derivative is \( f'(x) = \frac{-2}{3-2x} \).

Step by step solution

01

Understand the Problem

We need to find the derivative of the function given as \( f(x) = \ln |3-2x| \). This will involve using the rules of differentiation for logarithmic functions as well as the chain rule.
02

Differentiate the Logarithmic Function

The derivative of \( \ln |u| \) with respect to \( u \) is \( \frac{1}{u} \). For \( f(x) = \ln |3-2x| \), the inner function \( u \) is \( 3-2x \).
03

Apply the Chain Rule

The chain rule tells us that if \( y = g(u) \) and \( u = h(x) \), then \( \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} \). Here, \( g(u) = \ln |u| \) and \( h(x) = 3-2x \). So we need to find \( \frac{d}{dx}(3-2x) \).
04

Differentiate the Inner Function

The derivative of \( 3-2x \) with respect to \( x \) is \( -2 \).
05

Combine the Derivatives

Using the chain rule, \( f'(x) = \frac{1}{3-2x} \times (-2) \). Therefore, the derivative is \( f'(x) = \frac{-2}{3-2x} \).

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Key Concepts

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

Chain Rule
The chain rule is a fundamental technique in calculus used to differentiate composite functions. A composite function is simply a function within another function. To apply the chain rule, you need to:
  • Identify the outer and inner functions.
  • Differentiate the outer function with respect to the inner function.
  • Multiply the result by the derivative of the inner function.
For instance, in our given function \( f(x) = \ln |3-2x| \), the outer function is \( g(u) = \ln |u| \), and the inner function is \( u = 3 - 2x \). When we differentiate \( g(u) \) with respect to \( u \), we get \( \frac{1}{u} \). Then, we differentiate \( u = 3 - 2x \) with respect to \( x \), resulting in \( -2 \). The final derivative combines these results: \( f'(x) = \frac{1}{3-2x} \times (-2) = \frac{-2}{3-2x} \).
Differentiation
Differentiation is a process in calculus to find the rate at which a function changes at a given point. It is the core operation that allows us to calculate derivatives. In simpler words, it tells us how a function's output value changes as its input value changes slightly.

When differentiating, there are specific rules and methods we follow depending on the structure of the function:
  • Power Rule: For functions of the form \( x^n \), the derivative is \( nx^{n-1} \).
  • Product Rule: Used when differentiating the product of two functions.
  • Quotient Rule: Applied when differentiating a division of two functions.
  • Chain Rule: As discussed earlier, used for composite functions.
  • Exponential and Logarithmic Rules: Specific rules apply for exponential and logarithmic functions.
Understanding these rules and when to apply them is crucial in solving complex differentiation problems efficiently, such as the logarithmic function in this exercise.
Logarithmic Differentiation
Logarithmic differentiation is a technique useful for differentiating complex expressions and functions that involve products, quotients, or powers.

The steps to apply logarithmic differentiation include:
  • Take the natural logarithm of both sides of the equation \( y = f(x) \).
  • Use properties of logarithms to simplify the equation if possible.
  • Differentiate both sides with respect to \( x \) using implicit differentiation.
  • Solve for \( \frac{dy}{dx} \).
This method simplifies the differentiation process, especially with functions involving complex products or exponents. In our original example, we applied properties of logarithms along with the chain rule to determine \( f'(x) \). Such a technique can be very powerful when functions are too intricate to handle using standard methods.

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