Problem 14 - University Calculus: Early Transcendentals

Question

Differentiate the functions and find the slope of the tangent line at the given value of the independent variable. $$k(x)=\frac{1}{2+x}, \quad x=2$$

Step-by-Step Solution

Answer

The slope of the tangent line at \( x = 2 \) is \( \frac{-1}{16} \).

Step 1: Identify the Function and Variable

We have the function \( k(x) = \frac{1}{2+x} \) and we need to find its derivative to determine the slope of the tangent at \( x = 2 \).

Step 2: Use the Quotient Rule for Differentiation

The function \( k(x) = \frac{1}{2+x} \) can be differentiated using the quotient rule, \( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v\cdot u' - u\cdot v'}{v^2} \), where \( u = 1 \) and \( v = 2+x \). First, find the derivatives \( u' = 0 \) and \( v' = 1 \).

Step 3: Apply the Quotient Rule

Substitute the derivatives and functions into the quotient rule formula: \[ k'(x) = \frac{(2+x) \cdot 0 - 1 \cdot 1}{(2+x)^2} = \frac{-1}{(2+x)^2} \].

Step 4: Simplify the Derivative Expression

Simplify the derivative: \( k'(x) = \frac{-1}{(2+x)^2} \). This is the expression for the derivative of the function.

Step 5: Calculate the Derivative at \( x = 2 \)

Substitute \( x = 2 \) into \( k'(x) \) to find the slope. \[ k'(2) = \frac{-1}{(2+2)^2} = \frac{-1}{4^2} = \frac{-1}{16} \].

Step 6: Interpret the Result

The slope of the tangent line to the function at \( x = 2 \) is \( \frac{-1}{16} \). This means that the tangent line is slightly decreasing at \( x = 2 \).

Key Concepts

Quotient Rule

The Quotient Rule is an important technique in calculus used to differentiate functions that are written as a ratio of two functions, commonly denoted as \( \frac{u}{v} \). If you encounter a situation where one function is divided by another, you would use this rule to find the derivative or rate of change. The formula for the Quotient Rule is given by:

• \( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot u' - u \cdot v'}{v^2} \)

Here, \( u \) and \( v \) are both functions of \( x \) and \( u' \) and \( v' \) are their respective derivatives. This rule is particularly useful because it provides a systematic method to handle division of two differentiable functions.

A common mistake when using the Quotient Rule is forgetting to square the denominator. Always ensure that the entire denominator, \( v^2 \), is squared as shown in the formula. When calculating, keep track of each step and attribute the correct derivatives to \( u \) and \( v \). For example, in the function \( k(x) = \frac{1}{2+x} \), \( u \) equals 1 and \( v \) equals \( 2+x \). Thus, \( u' \) is 0 because the derivative of a constant is zero, and \( v' \) is 1. After substitution, you'll find that the derivative of \( k(x) \) simplifies to \( \frac{-1}{(2+x)^2} \).

Tangent Line Slope

The slope of a tangent line to a curve at a given point is essentially the derivative of the function at that specific point. It tells us how steep the curve is and whether it is increasing or decreasing. If you imagine the curve as a winding road, the tangent line represents a straight path that 'touches' this road at just one point without crossing it.

Calculating the tangent line slope involves two main steps:

• Differentiate the function to find its derivative.
• Substitute the given value of the independent variable (e.g., \( x = 2 \)) into the derivative to find the slope at that point.

In our provided exercise, the function \( k(x) = \frac{1}{2+x} \) has its derivative found to be \( \frac{-1}{(2+x)^2} \). By substituting \( x = 2 \) into this expression, we compute the slope as \( \frac{-1}{16} \). A negative slope value indicates that the tangent line is slightly decreasing at \( x = 2 \), meaning the curve is gently sloping downwards at that point.

Differentiation

Differentiation is a fundamental concept in calculus that helps us determine how a function changes at any given point. It's used to find the instantaneous rate of change, or in simpler terms, the slope of a curve at a particular point. This process is akin to answering the question: 'How fast is something changing at this exact moment?'

To differentiate a function, you can use various rules such as the Power Rule, Product Rule, or in this case, the Quotient Rule. Differentiation transforms a function into its derivative, offering insight into the function's behavior.

• It finds the rate of change of a dependent variable with respect to an independent variable.
• It helps to identify points where the function is increasing or decreasing.
• It calculates slopes of tangent lines, supporting curve sketching and modeling.

In the example of the function \( k(x) = \frac{1}{2+x} \), differentiation was applied using the Quotient Rule to find the derivative expression \( \frac{-1}{(2+x)^2} \). This derivative function is then used for further analysis, like determining the slope at \( x = 2 \). Differentiation is essential for many areas of study in mathematics, physics, engineering and economics.