Problem 14 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

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 1: 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 \(g(x) = x\).

Step 2: 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 \(g(x) = x\): $$g'(x) = 1$$

Step 3: Apply the Quotient Rule

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

Step 4: 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: $$\frac{d}{d x}\left(\frac{\ln x^{2}}{x}\right) = \frac{2 - \ln x^2}{x^2}$$

Key Concepts

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’t 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.