Problem 75 - Calculus for Biology and Medicine

Question

Find the first and the second derivatives of each function. \(g(x)=\frac{x-1}{x+1}\)

Step-by-Step Solution

Answer

The first derivative is \( g'(x) = \frac{2}{(x+1)^2} \) and the second derivative is \( g''(x) = \frac{-4}{(x+1)^3} \).

Step 1: Identify the Form of the Function

The function given is a rational function, meaning it is a ratio of two polynomials: the numerator is \(x - 1\) and the denominator is \(x + 1\). To find the derivatives, we'll need to apply the quotient rule.

Step 2: Recall the Quotient Rule for Differentiation

The quotient rule states that for two differentiable functions \( u(x) \) and \( v(x) \), the derivative of their quotient \( \frac{u(x)}{v(x)} \) is given by \( \frac{v(x)u'(x) - u(x)v'(x)}{[v(x)]^2} \). For our function, let \( u(x) = x-1 \) and \( v(x) = x+1 \).

Step 3: Differentiate the Numerator and Denominator

Differentiate \( u(x) = x - 1 \) to get \( u'(x) = 1 \). Differentiate \( v(x) = x + 1 \) to get \( v'(x) = 1 \).

Step 4: Apply the Quotient Rule

Apply the quotient rule: \[g'(x) = \frac{(x+1)(1) - (x-1)(1)}{(x+1)^2} = \frac{x + 1 - x + 1}{(x+1)^2}\]Simplify the expression: \[g'(x) = \frac{2}{(x+1)^2}\]

Step 5: Differentiate the First Derivative to Find the Second Derivative

The first derivative is \( g'(x) = \frac{2}{(x+1)^2} \). This can be expressed as \( 2(x+1)^{-2} \) for easier differentiation using the chain rule. Differentiate using the power rule along with the chain rule: \[g''(x) = 2 \cdot (-2)(x+1)^{-3} \cdot 1 = -4(x+1)^{-3}\]Simplify to write it back in fraction form: \[g''(x) = \frac{-4}{(x+1)^3}\]

Step 6: Summarize the Derivatives

The first derivative of the function \( g(x) = \frac{x-1}{x+1} \) is \( g'(x) = \frac{2}{(x+1)^2} \). The second derivative is \( g''(x) = \frac{-4}{(x+1)^3} \).

Key Concepts

Quotient Rule

The quotient rule is a fundamental technique in calculus used to differentiate ratios of two functions. It is applicable when we have a function expressed as a division between two differentiable functions. This rule is especially handy when dealing with rational functions, which are essentially divisions of polynomials.When applying the quotient rule, the formula we use is: \[\left( \frac{u}{v} \right)' = \frac{v \cdot u' - u \cdot v'}{v^2}\]where:

• \(u\) and \(v\) are differentiable functions.
• \(u'\) and \(v'\) are the derivatives of \(u\) and \(v\) respectively.

For the function \(g(x) = \frac{x-1}{x+1}\),

• Numerator \(u(x) = x-1\) with \(u'(x) = 1\).
• Denominator \(v(x) = x+1\) with \(v'(x) = 1\).

Applying the quotient rule gives us the first derivative:\[g'(x) = \frac{(x+1)(1) - (x-1)(1)}{(x+1)^2} = \frac{2}{(x+1)^2}\]

Chain Rule

The chain rule is a powerful tool in calculus for finding the derivative of composite functions. A composite function is where one function is applied inside another, such as \(f(g(x))\).The chain rule states:\[(f(g(x)))' = f'(g(x)) \cdot g'(x)\]This means you differentiate the outer function and then multiply by the derivative of the inner function. The chain rule is essential when differentiating expressions formed by power functions or nested functions.In the problem, to find the second derivative of \(g(x)\), we rewrite the first derivative \(g'(x)\) as \(2(x+1)^{-2}\). This allows us to use the chain rule to take the derivative once more:

• Differentiate the outer function \((x+1)^{-2}\), yielding \(-2(x+1)^{-3}\).
• The inner function \(x+1\) has a derivative of \(1\).

The second derivative thus becomes:\[g''(x) = 2 \cdot (-2)(x+1)^{-3} \cdot 1 = -4(x+1)^{-3}\]Transforming back to fraction form, the second derivative is:\[g''(x) = \frac{-4}{(x+1)^3}\]

Rational Function

A rational function is a type of function expressed as the ratio of two polynomials. These functions are characterized by a numerator and a denominator, both of which are polynomial functions. They are a significant part of algebra and calculus because they exhibit unique properties and behaviors.The general form of a rational function is:\[R(x) = \frac{P(x)}{Q(x)}\]where:

• \(P(x)\) is the numerator polynomial.
• \(Q(x)\) is the denominator polynomial and cannot be zero.

In our exercise, the rational function \(g(x) = \frac{x - 1}{x + 1}\) is composed of the polynomial \(x - 1\) over the polynomial \(x + 1\).Rational functions have interesting features:

• **Vertical asymptotes** where the denominator goes to zero.
• They can have horizontal asymptotes or oblique asymptotes, depending on the degrees of the polynomials.

Understanding the structure of rational functions is crucial for applying rules like the quotient rule for differentiation, as it allows us to handle both parts effectively when finding derivatives.