Problem 49 - Calculus for Biology and Medicine

Question

Differentiate the functions with respect to the independent variable. $$ f(x)=\frac{\ln x}{x} $$

Step-by-Step Solution

Answer

The derivative of \( f(x)=\frac{\ln x}{x} \) is \( f'(x) = \frac{1 - \ln x}{x^2} \).

Step 1: Identify the Appropriate Differentiation Technique

To differentiate the function \( f(x) = \frac{\ln x}{x} \), recognize that it has the form of a quotient, where \( u(x) = \ln x \) is the numerator and \( v(x) = x \) is the denominator. This suggests using the quotient rule, which is given by \( \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \).

Step 2: Differentiate the Numerator (\( u \))

Calculate the derivative of the numerator function \( u(x) = \ln x \). The derivative of \( \ln x \) with respect to \( x \) is \( \frac{1}{x} \). Therefore, \( u'(x) = \frac{1}{x} \).

Step 3: Differentiate the Denominator (\( v \))

Calculate the derivative of the denominator function \( v(x) = x \). The derivative of \( x \) with respect to \( x \) is \( 1 \). Thus, \( v'(x) = 1 \).

Step 4: Apply the Quotient Rule

Substitute \( u' = \frac{1}{x} \), \( v = x \), \( u = \ln x \), and \( v' = 1 \) into the quotient rule formula: \( \left( \frac{\ln x}{x} \right)' = \frac{\left( \frac{1}{x} \right)x - (\ln x)(1)}{x^2} \). Simplifying this expression gives us the derivative.

Step 5: Simplify the Result

First, simplify the expression from Step 4: \( \frac{1}{x} \cdot x - \ln x = 1 - \ln x \). Thus, the derivative is \( \frac{1 - \ln x}{x^2} \).

Step 6: State the Final Derivative

The derivative of the function \( f(x) = \frac{\ln x}{x} \) with respect to \( x \) is given by \( f'(x) = \frac{1 - \ln x}{x^2} \). This represents the rate of change of \( f(x) \) with respect to \( x \).

Key Concepts

Quotient Rule in Differentiation

When you have a function expressed as a ratio of two differentiable functions, like \(\frac{u(x)}{v(x)}\), the quotient rule is your go-to tool for differentiation. It is particularly handy in breaking down complex-looking problems involving fractions. When differentiating such a quotient, it’s vital to remember the formula: \[ \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \] This rule tells us how to differentiate the numerator, \(u(x)\), and the denominator, \(v(x)\), separately. Then you plug these values back into the formula. Break it down by:

• Finding \(u'\) and \(v'\): First focus on differentiating \(u(x)\) and \(v(x)\) individually.
• Substituting values: Insert these derivatives into the formula:

This organized approach helps keep your work error-free and understandable. In our case, the function \(\frac{\ln x}{x}\) fits perfectly into the rule, making the differentiation simpler.

Understanding the Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\), is a fundamental concept in calculus and algebra. It is based on the number \(e\) (approximately 2.71828), which is the base of the natural logarithm. The function \(\ln(x)\) answers the question, "To what power must \(e\) be raised to yield \(x\)?"

• The natural logarithm is the inverse of the exponential function with base \(e\).
• If \(y = \ln(x)\), then the equivalent exponential form is \(x = e^y\).

When differentiating \(\ln(x)\), its derivative is given by \(\frac{1}{x}\). This formula is straightforward yet crucial when solving problems that involve products or quotients, such as the one we have addressed here. Understanding \(\ln(x)\) helps in recognizing how different functions interact, particularly in real-world applications, including compound interest and population growth.

The Basics of Derivatives

Derivatives are at the heart of differential calculus and are used to represent how a function changes as its input changes. The derivative of a function at any given point describes the slope of the tangent line to the function at that point. In simple terms, it tells us the rate of change of one variable with respect to another. For example, the derivative \(f'(x)\) provides the rate at which \(f(x)\) changes with an incremental change in \(x\).

• Understanding the Concept: Derivatives reveal trends and tendencies in data, predicting behavior with changes in variables.
• Notations: The derivative of a function \(f(x)\) can be denoted as \(f'(x)\), \(\frac{df}{dx}\), or \(D_x[f]\).
• Basic Rules: Familiarity with rules like the power rule, product rule, and quotient rule significantly simplifies the differentiation process.

In the exercise provided, knowing how to correctly apply the quotient rule empowers you to efficiently manage and simplify the function differentiation. The power of derivatives extends far beyond classroom problems, impacting fields like physics, engineering, economics, and beyond, making it a cornerstone in mathematical analysis.