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Chapter 4: Problem 93

Choose the correct answer: \(\frac{d}{d x} \ln x=\) a. \(\frac{x}{1}\) b. \(\frac{1}{x}\) c. 0

Short Answer

Expert verified
The correct answer is b: \(\frac{1}{x}\).

Step by step solution

01

Understand the Function

We are given the function \(\ln x\), which refers to the natural logarithm of \(x\). Our goal is to compute the derivative of this function with respect to \(x\).
02

Recall the Derivative Rule for Natural Logarithm

Recall that the derivative of \(\ln x\) is a well-known derivative rule in calculus, which states \(\frac{d}{dx} [\ln x] = \frac{1}{x}\).
03

Compute the Derivative

Apply the derivative rule from Step 2 to the function \(\ln x\). The derivative is computed as \(\frac{d}{dx} [\ln x] = \frac{1}{x}\).
04

Choose the Correct Answer

Compare the computed derivative \(\frac{1}{x}\) with the provided options. The correct choice is option b: \(\frac{1}{x}\).

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Key Concepts

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

Natural Logarithm
The natural logarithm is a specific logarithmic function where the base of the logarithm is Euler's number, denoted as \(e\). The number \(e\) is approximately equal to 2.71828 and is an irrational number.

The natural logarithm of a positive number \(x\) is written as \(\ln x\).
  • Natural logs are used to transform multiplicative processes into additive processes, which can simplify calculations in various mathematical applications.
  • The natural logarithm is the inverse function of the exponential function with the base \(e\); this means if \(y = \ln x\), then \(x = e^y\).
Understanding \(\ln x\) as a function is important in calculus, particularly when we need to perform operations such as differentiation.
Derivative Rules
In calculus, derivative rules are systematic ways to find the derivative of functions. These rules simplify the process of differentiation, allowing us to quickly find the rate at which functions change.
Some common derivative rules include:
  • Power Rule: For \(f(x) = x^n\), the derivative \(f'(x) = nx^{n-1}\).
  • Product Rule: For two functions \(f(x)\) and \(g(x)\), the derivative \((fg)'(x) = f'(x)g(x) + f(x)g'(x)\).
  • Quotient Rule: For two functions \(f(x)\) and \(g(x)\), the derivative \(\left(\frac{f}{g}\right)'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}\).
  • Chain Rule: For the composition of two functions \(f(g(x))\), the derivative \(f'(g(x))g'(x)\).
The derivative of the natural logarithm: \(\frac{d}{dx} [\ln x] = \frac{1}{x}\), is another essential derivative rule, specifically dealing with logarithmic functions.
Differentiation
Differentiation is the fundamental operation in calculus used to find the derivative of a function. It measures how a function changes as its input changes, effectively providing the rate of change or slope of the function at any given point.
There are several important aspects of differentiation:
  • Differentiation allows for the analysis of real-world phenomena, such as velocity and acceleration, by dealing with changing quantities.
  • It helps in finding the maxima and minima of functions, useful in optimization problems.
  • Different methods and rules can be applied in differentiation, including the rules for polynomial, logarithmic, and trigonometrical functions.
By understanding differentiation, and the specific derivatives such as that of the natural logarithm, we can deal with a wide range of mathematical problems more efficiently.

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Most popular questions from this chapter

For each demand function \(D(p)\) : a. Find the elasticity of demand \(E(p)\). b. Determine whether the demand is elastic, inelastic, or unit-elastic at the given price \(p\). $$ D(p)=\frac{600}{p^{3}}, \quad p=25 $$

For each demand function \(D(p)\) : a. Find the elasticity of demand \(E(p)\). b. Determine whether the demand is elastic, inelastic, or unit-elastic at the given price \(p\). $$ D(p)=100-p^{2}, p=5 $$

An automobile dealer is selling cars at a price of $$\$ 12,000$$. The demand function is \(D(p)=2(15-0.001 p)^{2}\), where \(p\) is the price of a car. Should the dealer raise or lower the price to increase revenue?

For each function: a. Find the relative rate of change. b. Evaluate the relative rate of change at the given value(s) of \(t\). $$ f(t)=e^{-t^{2}}, \quad t=10 $$

A supply function \(S(p)\) gives the total amount of a product that producers are willing to supply at a given price \(p\). The elasticity of supply is defined as $$ E_{s}(p)=\frac{p \cdot S^{\prime}(p)}{S(p)} $$ Elasticity of supply measures the relative increase in supply resulting from a small relative increase in price. It is less useful than elasticity of demand, however, since it is not related to total revenue. Use the preceding formula to find the elasticity of supply for a supply function of the form \(S(p)=a e^{c p}\), where \(a\) and \(c\) are positive constants.
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