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Chapter 3: Problem 14

Find the following derivatives. $$\frac{d}{d x}\left(\frac{\ln x^{2}}{x}\right)$$

Short Answer

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Question: Find the derivative of the function \(\frac{\ln x^2}{x}\). Answer: The derivative of the function is \(\frac{d}{d x}\left(\frac{\ln x^{2}}{x}\right) = \frac{2 - \ln x^2}{x^2}\).

Step by step solution

01

Recall the Quotient Rule

The Quotient Rule for finding the derivative of a fraction of two functions is as follows: $$\left(\frac{f(x)}{g(x)}\right)' = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$$ Here, \(f(x) = \ln x^2\) and \(g(x) = x\).
02

Find derivatives of the numerator and denominator

First, we find the derivative of \(f(x) = \ln x^2\). Using the Chain Rule, we have: $$f'(x) = \frac{\partial}{\partial x^2} (\ln x^2) \frac{\partial x^2}{\partial x} = \frac{1}{x^2} (2x) = \frac{2}{x}$$ Now, we find the derivative of \(g(x) = x\): $$g'(x) = 1$$
03

Apply the Quotient Rule

Now that we have \(f'(x)\) and \(g'(x)\), we apply the quotient rule: $$\left(\frac{\ln x^{2}}{x}\right)' = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2} = \frac{\frac{2}{x} \cdot x - \ln x^2 \cdot 1}{x^2}$$
04

Simplify the expression

Next, we simplify the above expression: $$\left(\frac{\ln x^{2}}{x}\right)' = \frac{2 - \ln x^2}{x^2}$$ So, the derivative of the function is: $$\frac{d}{d x}\left(\frac{\ln x^{2}}{x}\right) = \frac{2 - \ln x^2}{x^2}$$

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Key Concepts

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

Quotient Rule
The Quotient Rule is a technique used to find the derivative of a function that is the division of two other functions. Since it deals with fractions, it's especially handy when you can't easily simplify a function before differentiating. The rule is applied as follows: if you have a fraction of functions \(\frac{f(x)}{g(x)}\),
  • first, find the derivative of the top function, \(f'(x)\);
  • then, find the derivative of the bottom function, \(g'(x)\);
  • the derivative of the entire fraction is given by the formula: \[ \left(\frac{f(x)}{g(x)}\right)' = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \]
After applying this rule, simplifying the result is important to obtain the cleanest expression possible. In this exercise, we used the Quotient Rule to differentiate \(\frac{\ln x^2}{x}\), leading us to the final derivative \(\frac{2 - \ln x^2}{x^2}\).
Chain Rule
The Chain Rule is an excellent tool for differentiating composite functions, which are functions composed of an inside and an outside function. In essence, it lets us "chain" together derivatives of nested functions to find our result. For a composite function \(f(g(x))\), the Chain Rule states:
  • first, differentiate the outer function \(f\) with respect to the inner function \(g(x)\), often written as \(f'(g(x))\);
  • then multiply this result by the derivative of the inner function \(g'(x)\).
This results in the formula: \[ \frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x) \] In our exercise, the Chain Rule was crucial in differentiating \(\ln x^2\), where the outer function is \(\ln\) and the inner function is \(x^2\). We derived \(\frac{2}{x}\) by chaining together these derivatives.
Logarithmic Differentiation
Logarithmic differentiation is a powerful method, particularly when dealing with complex products or quotients, as well as composition of functions involving logs. It involves taking the natural logarithm on both sides of an equation before differentiating.
  • First, apply the log to simplify the expression; this often turns products and powers into sums and manageable derivatives.
  • Then, differentiate both sides with respect to \(x\).
  • Finally, solve for the derivative you are seeking.
In the given problem, although we didn鈥檛 explicitly go through full logarithmic differentiation, utilizing the function \(\ln x^2\) relates to it. By recognizing \(\ln x^2 = 2\ln x\), we simplify differentiation processes and connect back to the method when derivatives of log functions appear. It's a neat technique to keep in your arsenal when tackling functions where direct differentiation seems daunting.

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Most popular questions from this chapter

a. Determine an equation of the tangent line and normal line at the given point \(\left(x_{0}, y_{0}\right)\) on the following curves. b. Graph the tangent and normal lines on the given graph. \(x^{4}=2 x^{2}+2 y^{2}\) \(\left(x_{0}, y_{0}\right)=(2,2)\) (kampyle of Eudoxus)

Suppose a large company makes 25,000 gadgets per year in batches of \(x\) items at a time. After analyzing setup costs to produce each batch and taking into account storage costs, it has been determined that the total cost \(C(x)\) of producing 25,000 gadgets in batches of \(x\) items at a time is given by $$C(x)=1,250,000+\frac{125,000,000}{x}+1.5 x.$$ a. Determine the marginal cost and average cost functions. Graph and interpret these functions. b. Determine the average cost and marginal cost when \(x=5000\). c. The meaning of average cost and marginal cost here is different from earlier examples and exercises. Interpret the meaning of your answer in part (b).

The flow of a small stream is monitored for 90 days between May 1 and August 1. The total water that flows past a gauging station is given by $$V(t)=\left\\{\begin{array}{ll}\frac{4}{5} t^{2} & \text { if } 0 \leq t<45 \\\\-\frac{4}{5}\left(t^{2}-180 t+4050\right) & \text { if } 45 \leq t<90, \end{array}\right.$$ where \(V\) is measured in cubic feet and \(t\) is measured in days, with \(t=0\) corresponding to May 1. a. Graph the volume function. b. Find the flow rate function \(V^{\prime}(t)\) and graph it. What are the units of the flow rate? c. Describe the flow of the stream over the 3 -month period. Specifically, when is the flow rate a maximum?

Quotient Rule for the second derivative Assuming the first and second derivatives of \(f\) and \(g\) exist at \(x,\) find a formula for \(\frac{d^{2}}{d x^{2}}\left(\frac{f(x)}{g(x)}\right)\)

A 500-liter (L) tank is filled with pure water. At time \(t=0,\) a salt solution begins flowing into the tank at a rate of \(5 \mathrm{L} / \mathrm{min} .\) At the same time, the (fully mixed) solution flows out of the tank at a rate of \(5.5 \mathrm{L} / \mathrm{min}\). The mass of salt in grams in the tank at any time \(t \geq 0\) is given by $$M(t)=250(1000-t)\left(1-10^{-30}(1000-t)^{10}\right)$$ and the volume of solution in the tank (in liters) is given by \(V(t)=500-0.5 t\). a. Graph the mass function and verify that \(M(0)=0\). b. Graph the volume function and verify that the tank is empty when \(t=1000\) min. c. The concentration of the salt solution in the tank (in \(\mathrm{g} / \mathrm{L}\) ) is given by \(C(t)=M(t) / V(t) .\) Graph the concentration function and comment on its properties. Specifically, what are \(C(0)\) and \(\lim _{\theta \rightarrow 000^{-}} C(t) ?\) \(t \rightarrow 1\) d. Find the rate of change of the mass \(M^{\prime}(t),\) for \(0 \leq t \leq 1000\). e. Find the rate of change of the concentration \(C^{\prime}(t),\) for \(0 \leq t \leq 1000\). f. For what times is the concentration of the solution increasing? Decreasing?
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