Problem 57 - Calculus for Biology and Medicine

Question

Differentiate with respect to the independent variable. $$ f(s)=\frac{4-2 s^{2}}{1-s} $$

Step-by-Step Solution

Answer

The derivative is \( f'(s) = \frac{2s^2 - 4s + 4}{(1-s)^2} \).

Step 1: Identify the Function Type

The given function \(f(s) = \frac{4-2s^2}{1-s}\) is a rational function, which means it is a division of two polynomials.

Step 2: Recall the Quotient Rule

The derivative of a quotient \(\frac{u}{v}\) is given by the quotient rule: \(\frac{d}{ds}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}\), where \(u = 4 - 2s^2\) and \(v = 1 - s\).

Step 3: Differentiate the Numerator

Differentiate \(u = 4 - 2s^2\) with respect to \(s\). The derivative is \(u' = -4s\).

Step 4: Differentiate the Denominator

Differentiate \(v = 1 - s\) with respect to \(s\). The derivative is \(v' = -1\).

Step 5: Apply the Quotient Rule

Substitute \(u\), \(u'\), \(v\), and \(v'\) into the quotient rule to find the derivative: \[f'(s) = \frac{(-4s)(1-s) - (4-2s^2)(-1)}{(1-s)^2}.\]

Step 6: Simplify the Expression

Simplify the expression inside the numerator:\[f'(s) = \frac{-4s + 4s^2 + 4 - 2s^2}{(1-s)^2} = \frac{2s^2 - 4s + 4}{(1-s)^2}.\] This is the simplified form of the derivative of the function.

Key Concepts

Rational Functions

Rational functions are a type of function that is represented by the ratio of two polynomials. In simpler terms, it's a fraction where both the numerator and the denominator are polynomial expressions. For example, in this exercise, the given function is \[ f(s) = \frac{4 - 2s^2}{1 - s} \] Here, the numerator is a polynomial \(4 - 2s^2\) and the denominator is another polynomial \(1 - s\). This setup allows rational functions to exhibit a wide range of behavior, including vertical and horizontal asymptotes, and discontinuities at certain points. Because of the polynomial nature of both components, when dealing with rational functions, it's crucial to understand both how they behave and how to manipulate them mathematically, especially when finding derivatives.

Quotient Rule

The quotient rule is an essential tool in calculus used to differentiate functions that are expressed as the quotient of two other functions. If you have a function \(\frac{u}{v}\), where both \(u\) and \(v\) are differentiable functions, the quotient rule helps to differentiate it efficiently. The quotient rule formula is: \[ \frac{d}{ds}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2} \]This formula shows that you need to:

• Differentiate the numerator function \(u\), resulting in \(u'\).
• Differentiate the denominator function \(v\), which gives \(v'\).
• Apply these derivatives in the formula, keeping in mind to substitute the original functions as well.

Utilizing the quotient rule is crucial when working with expressions like the one in our exercise, as it provides a structured way to find the derivative of more complex functions that cannot be simplified easily before differentiating.

Derivatives

Derivatives measure how a function changes as its input changes. They can be thought of as a way to determine the rate of change or the slope of a function at any given point. For instance, in the derivation process provided, the differentiation of the numerator and the denominator of the rational function involves:

• For the numerator \(u = 4 - 2s^2\), the derivative \(u'\) is found by applying basic differentiation rules, yielding \(-4s\).
• For the denominator \(v = 1 - s\), the derivative \(v'\) is simply \(-1\).

These derivatives are then used in the quotient rule to find the overall derivative of the rational function. Understanding how to calculate derivatives is fundamental in calculus, as it's widely used in various applications such as physics, engineering, and economics to analyze change.

Simplifying Expressions

After applying rules such as the quotient rule to find a derivative, it's often necessary to simplify the resulting expression. Simplification makes the derivative easier to interpret, and sometimes more functional for further mathematical operations. In this exercise, after applying the quotient rule, the initial unsimplified derivative is:\[ f'(s) = \frac{(-4s)(1-s) - (4-2s^2)(-1)}{(1-s)^2} \]Simplifying involves distributing, combining like terms, and re-organizing the expression to a more concise form:\[ f'(s) = \frac{2s^2 - 4s + 4}{(1-s)^2} \]Key techniques used in simplifying include:

• Distributing multiplication over addition and subtraction within each part of the expression.
• Combining like terms to reduce complexity.
• Simplifying fractions and factoring where possible to ensure the expression is presented in its simplest form.

Simplification is an integral part of calculus as it not only provides clarity but also facilitates further computations and analysis.