Question

Compute the derivative of the given function. $$f(x)=2 \ln x-x$$

Short answer

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

Step-by-step solution

Identify the Form of the Function

The function given is of the form \( f(x) = 2 \ln x - x \). It is the difference between a logarithmic function and a linear function.

Recall the Derivative Rules

To find the derivative, we need the derivative rules for the functions involved. The derivative of \( \ln x \) is \( \frac{1}{x} \) and the derivative of \( x \) is 1. The derivative of a constant times a function is the constant times the derivative of the function. Lastly, the difference rule states that the derivative of a difference of functions is the difference of their derivatives.

Apply the Derivative to Each Term

Using the derivative rules, apply the derivative to each term of \( f(x) = 2 \ln x - x \): \[ f'(x) = \frac{d}{dx}(2 \ln x) - \frac{d}{dx}(x). \] Calculate each derivative separately next.

Differentiate the First Term

Differentiate \( 2 \ln x \): \[ \frac{d}{dx}(2 \ln x) = 2 \cdot \frac{d}{dx}(\ln x) = 2 \cdot \frac{1}{x} = \frac{2}{x}. \]

Differentiate the Second Term

Differentiate \( x \): \[ \frac{d}{dx}(x) = 1. \]

Combine the Results

Combine the derivatives obtained in Steps 4 and 5 using the difference rule: \[ f'(x) = \frac{2}{x} - 1. \] Thus, the derivative of the function \( f(x) = 2 \ln x - x \) is \( f'(x) = \frac{2}{x} - 1. \)

Key concepts

Logarithmic Function

A logarithmic function is a type of function that appears in the form of \( \ln x \, \). The "ln" stands for the natural logarithm, which is the logarithm to the base \( \mathrm{e}\ \), the irrational number approximately equal to 2.71828. Logarithmic functions are essentially the inverse operations of exponential functions. They are incredibly useful for solving equations where the variable appears as an exponent.
Logarithmic functions have unique properties:

• The function \( \ln x \, \) is only defined for \( x > 0 \).
• The product property: \( \ln (ab) = \ln a + \ln b \).
• The quotient property: \( \ln \left( \frac{a}{b} \right) = \ln a - \ln b \).
• The power property: \( \ln (a^b) = b \ln a \).

Understanding these properties can help simplify logarithmic expressions and make differentiation easier.
Logarithmic differentiation requires recalling that the derivative of \( \ln x \, \) is \( \frac{1}{x}\ \). This is a rule you'll use often when dealing with any function involving logarithmic terms.

Linear Function

A linear function is a straight-line graph, typically described in the form \( \ f(x) = mx + b \, \). Here, \( m \) is the slope of the line, and \( b \) is the y-intercept, or the point where the line crosses the y-axis.
**Key Characteristics of Linear Functions:**

• The graph is a straight line.
• Each unit increase in \( x \) results in a consistent increase or decrease in \( f(x) \), based on the slope.
• The derivative, or the rate of change, of a simple linear function \( x \) with respect to \( x \) is always 1.

In the function \( \ f(x) = 2 \ln x - x\ \), the term \( -x \) is a linear function. Its derivative is straightforward, \( \ -1 \), much simpler compared to other complex functions. Understanding linear functions and their derivatives provides foundational insight to tackle more complicated composites of functions.

Difference Rule

The difference rule in calculus is as simple as it sounds. It states that to differentiate the difference of two functions, you can take the derivatives of each function separately and then subtract one from the other.
Mathematically, if you have two functions \( \ u(x) \) and \( v(x) \, \), and you need to find the derivative of \( \ u(x) - v(x) \, \), then:\[ \frac{d}{dx}\left [ u(x) - v(x) \right ] = u'(x) - v'(x).\]This is useful when you work on equations with multiple components, like linear and logarithmic combinations.
In our example \( \ f(x) = 2 \ln x - x \, \), the difference rule allows us to apply the derivative rules separately on \( 2 \ln x \, \) and \( -x \, \) and then combine the results to get \( \ f'(x) = \frac{2}{x} - 1 \. \) This simplicity underscores one of calculus's key strengths—breaking complicated problems into smaller, manageable parts.

Derivative Rules

Derivative rules are essential tools in calculus that guide us in finding the derivative of a function, which represents its rate of change. Several important rules include the power rule, constant multiple rule, sum/difference rule, and chain rule.
**Here are some fundamental derivative rules:**

• Power Rule: For \( \ f(x) = x^n \, \), \( \ f'(x) = nx^{n-1} \. \)
• Constant Multiple Rule: If \( \ c \) is a constant, \( \ d( [cf(x)] / dx ) = c f'(x) \. \)
• Difference Rule: For \( \ u(x) - v(x) \, \), \( \ \frac{d}{dx}[u(x)-v(x)] = u'(x) - v'(x) \. \)

In the context of \( \ f(x) = 2 \ln x - x \, \)
- The constant multiple rule handles the \( \ 2 \ln x \, \) giving \( \ \frac{2}{x} \ \).
- The linear function \( -x \, \) uses the power rule, resulting in \( -1 \).
By combining these rules, we efficiently find that \( f'(x) = \frac{2}{x} - 1 \).