/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 16 In Exercises \(1-28\) , find \(d... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 16

In Exercises \(1-28\) , find \(d y / d x\) . Remember that you can use NDER to support your computations. $$y=(\ln x)^{2}$$

Short Answer

Expert verified
The derivative of the function \(y=(\ln x)^{2}\) with respect to \(x\) is \(dy/dx = 2*ln(x)/x\).

Step by step solution

01

Identify the outer and inner functions

In the chain rule, if a function \(y\) can be written in the form \(y=f(g(x))\), then \(y' = f'(g(x))*g'(x)\). For the given function \(y=(\ln x)^{2}\), the outer function is \(f(u) = u^2\) and the inner function is \(g(x) = \ln x\).
02

Differentiate Outer and Inner Functions

Apply the power rule to the outer function \(f(u) = u^2\), causing the derivative to become \(f'(u) = 2u\). Differentiate the inner function \(g(x) = \ln x\) to obtain \(g'(x) = 1/x\).
03

Apply Chain Rule

Now apply the chain rule which states that if a variable \(z\) depends on the variable \(y\), which itself depends on the variable \(x\), then \(z\), via the intermediate variable of \(y\), depends on \(x\) as: \(dz/dx = dz/dy * dy/dx\). Substitute the derivatives of the outer and inner functions, so the derivative \(dy/dx\) is \(f'(g(x))*g'(x) = 2*ln x*(1/x) = 2*ln(x)/x\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Logarithmic Differentiation
Logarithmic differentiation offers an alternative approach to finding the derivative of functions that are difficult to differentiate using standard rules. This is particularly useful when dealing with functions raised to powers, variables in exponents, or products and quotients of functions.

The process involves taking the natural logarithm of both sides of an equation and then differentiating implicitly. This approach simplifies the differentiation process by turning multiplication into addition and division into subtraction, according to the properties of logarithms.
  • For multiplication, \( \ln(a \cdot b) = \ln(a) + \ln(b) \)
  • For division, \( \ln(\frac{a}{b}) = \ln(a) - \ln(b) \)
After taking the logarithm, one can apply the derivative of the natural logarithm along with the chain rule to find the derivative of complicated functions.
Power Rule
The power rule is one of the most fundamental rules of differentiation and is used to find the derivative of a function in the form of \(f(x) = x^n\), where \(n\) is any real number.

The rule states that if \(f(x) = x^n\), then the derivative \(f'(x) = n \cdot x^{n-1}\). This is an immediate consequence of the definition of the derivative and provides a quick and efficient way to differentiate functions with polynomial terms.

In the exercise, we apply the power rule to \(f(u) = u^2\), where \(u = \ln(x)\). The power rule simplifies this to \(2u\), which we then multiply by the derivative of \(u\), using the chain rule.
Derivative of Natural Logarithm
The natural logarithm function, denoted by \(\ln(x)\), has a simple but incredibly important derivative in calculus. The derivative of \(\ln(x)\) with respect to \(x\) is \(\frac{1}{x}\). This rule springs directly from the definition of the derivative and the special properties of the natural logarithm.

When differentiating functions involving \(\ln(x)\), it's important to remember that its derivative only applies directly to functions of the form \(\ln(x)\) and must be adjusted using the chain rule when dealing with a composite function like \(\ln(g(x))\).

In our example, the inner function \(g(x) = \ln x\) is differentiated to \(g'(x) = \frac{1}{x}\), and then this derivative is used in conjunction with the outer function's derivative via the chain rule to calculate the overall derivative of the function.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use analytic methods to show that \(\lim _{h \rightarrow 0} \frac{\cos h-1}{h}=0\)

In Exercises 61 and \(62,\) use the curve \(x^{2}-x y+y^{2}=1\) Multiple Choice Which of the following is equal to \(d y / d x ?\) (A) \(\frac{y-2 x}{2 y-x} \quad\) (B) \(\frac{y+2 x}{2 y-x}\) (C) \(\frac{2 x}{x-2 y} \quad\) (D) \(\frac{2 x+y}{x-2 y}\) \((\mathbf{E}) \frac{y+2 x}{x}\)

Find an equation for a line that is tangent to the graph of \(y=e^{x}\)and goes through the origin.

Generating the Birthday Probabilities Example 5 of this section concerns the probability that, in a group of \(n\) people, at least two people will share a common birthday. You can generate these probabilities on your calculator for values of \(n\) from 1 to \(365 .\) Step 1: Set the values of \(N\) and \(P\) to zero: Step \(2 :\) Type in this single, multi-step command: Now each time you press the ENTER key, the command will print a new value of \(N(\) the number of people in the room) alongside \(P\) (the probability that at least two of them share a common birthday): If you have some experience with probability, try to answer the following questions without looking at the table: (a) If there are three people in the room, what is the probability that they all have different birthdays? (Assume that there are 365 possible birthdays, all of them equally likely.) (b) If there are three people in the room, what is the probability that at least two of them share a common birthday? (c) Explain how you can use the answer in part (b) to find the probability of a shared birthday when there are four people in the room. (This is how the calculator statement in Step 2 generates the probabilities.) (d) Is it reasonable to assume that all calendar dates are equally likely birthdays? Explain your answer.

Multiple Choice Which of the following is the domain of \(f^{\prime}(x)\) if \(f(x)=\log _{2}(x+3) ? \quad\) (A) \(x<-3 \quad\) (B) \(x \leq 3 \quad\) (C) \(x \neq-3 \quad\) (D) \(x>-3\) (E) \(x \geq-3\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.