Problem 300 - Calculus Volume 1

Question

For the following exercises, find the derivative dy/dx. (You can use a calculator to plot the function and the derivative to confirm that it is correct.) [T] \(y=\frac{\ln (x)}{x}\)

Step-by-Step Solution

Answer

\( \frac{dy}{dx} = \frac{1 - \ln(x)}{x^2} \).

Step 1: Identify the Rule

The given function is a quotient of functions. To find the derivative \( \frac{dy}{dx} \), use the Quotient Rule. The Quotient Rule is \( \frac{d}{dx}\left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2} \), where \( u \) is the numerator function and \( v \) is the denominator function.

Step 2: Define u and v

Let \( u = \ln(x) \) and \( v = x \). Identify the derivatives \( u' \) and \( v' \).

Step 3: Differentiate u

Find the derivative of \( u \):\[u' = \frac{d}{dx}(\ln(x)) = \frac{1}{x}.\]

Step 4: Differentiate v

Find the derivative of \( v \):\[v' = \frac{d}{dx}(x) = 1.\]

Step 5: Apply the Quotient Rule

Substitute \( u = \ln(x), u' = \frac{1}{x}, v = x, \) and \( v' = 1 \) into the Quotient Rule formula:\[\frac{dy}{dx} = \frac{\left(\frac{1}{x}\right) \cdot x - (\ln(x)) \cdot 1}{x^2}.\]

Step 6: Simplify the Expression

Simplify the expression:\[\frac{dy}{dx} = \frac{1 - \ln(x)}{x^2}.\] This is the derivative of \( y = \frac{\ln(x)}{x} \).

Key Concepts

Understanding the Quotient Rule

In calculus, the quotient rule is a method for finding the derivative of a ratio of two differentiable functions. When you have a function expressed as a fraction, the quotient rule becomes very handy. It helps differentiate functions that are divided by one another. Simply put, when you want to find the derivative of a function of the form \( \frac{u}{v} \), the quotient rule states that:

• Take the derivative of the numerator function \( u \), which we'll call \( u' \).
• Take the derivative of the denominator function \( v \), referred to as \( v' \).
• Apply the formula: \[ \frac{dy}{dx} = \frac{u'v - uv'}{v^2}. \]

This formula shows that you multiply the derivative of the numerator by the denominator, subtract the product of the numerator and the derivative of the denominator, and then divide everything by the square of the denominator. It may sound complex at first, but with practice, this process becomes second nature.

Exploring Logarithmic Functions

Logarithmic functions are a class of functions that involve the natural logarithm, often represented by \( \ln(x) \). The natural logarithm is the inverse of the exponential function, and it's crucial when dealing with both growth and decay processes in science and engineering. A key property of logarithmic functions is that their derivative is straightforward to find, specifically for \( \ln(x) \):

• The derivative is \( \frac{1}{x} \).

This derivation is essential because it helps in simplifying the application of the quotient rule. In the context of our original problem, the function \( y = \frac{\ln(x)}{x} \) clearly shows the logarithmic component in the numerator. Recognizing logarithmic functions and knowing their derivatives can greatly simplify the process of finding the derivative using the quotient rule. When you see a \( \ln(x) \), always remember how simple its derivative is.

Step-by-Step Differentiation Process

Differentiation is the process of finding the derivative, or the rate at which a function is changing at any given point. Let’s break down the differentiation process for the function \( y = \frac{\ln(x)}{x} \) using the quotient rule:

• Step 1: Identify \( u \) and \( v \) from the function — \( u = \ln(x) \) and \( v = x \).
• Step 2: Differentiate \( u \): The derivative \( u' = \frac{1}{x} \).
• Step 3: Differentiate \( v \): The derivative \( v' = 1 \).
• Step 4: Plug these values into the quotient rule formula:\[\frac{dy}{dx} = \frac{\left(\frac{1}{x}\right) \cdot x - (\ln(x)) \cdot 1}{x^2}. \]
• Step 5: Simplify the result to get:\[\frac{dy}{dx} = \frac{1 - \ln(x)}{x^2}. \]

These steps help ensure that you are methodically working through the process to arrive at a simplified expression for the derivative. By following a structured approach, you minimize errors and increase your understanding of how the derivative relates to changes in the original function.