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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

Expert verified
Question: Find the derivative of the function $\frac{\ln x^{2}}{x}$ with respect to x. Answer: The derivative of the given function with respect to x is $\frac{d}{dx}(\frac{\ln x^{2}}{x}) = \frac{2 - \ln(x^{2})}{x^2}$.

Step by step solution

01

Identify the functions

We have a quotient of functions, where the numerator is the natural logarithm of x^2 and the denominator is x. Let the numerator be u(x) and the denominator be v(x). $$ u(x) = \ln(x^2) \\ v(x) = x $$
02

Differentiating u and v

In this step, we'll find the derivatives of u(x) and v(x) with respect to x, using the chain rule for the natural logarithm function and the power rule for x. $$ u'(x) = \frac{d}{dx}(\ln(x^2))=\frac{1}{x^2}\cdot\frac{d}{dx}(x^2)=\frac{1}{x^2}\cdot 2x = \frac{2}{x} \\ v'(x) = \frac{d}{dx}(x)=1 $$
03

Apply the quotient rule

The quotient rule states that if you have a function y(x) = u(x)/v(x), then its derivative is given by: $$ \frac{dy}{dx}=\frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} $$ Now, apply the quotient rule to the given function, with u(x) and v(x) and their derivatives from the previous steps: $$ \frac{d}{dx}\left(\frac{\ln x^{2}}{x}\right) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} = \frac{\frac{2}{x}\cdot x - \ln(x^{2})\cdot 1}{x^2} $$
04

Simplify the expression

Now, simply the expression obtained in the previous step: $$ \frac{d}{dx}\left(\frac{\ln x^{2}}{x}\right) = \frac{2 - \ln(x^{2})}{x^2} $$ The derivative of the given function with respect to x is: $$ \frac{d}{d x}\left(\frac{\ln x^{2}}{x}\right) = \frac{2 - \ln(x^{2})}{x^2} $$

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