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 Structure

The function given is a rational function, where the numerator is a quadratic expression in terms of \(s\) and the denominator is a linear expression in terms of \(s\). We have \(f(s) = \frac{4 - 2s^2}{1 - s}\).

Step 2: Apply the Quotient Rule

For a quotient \(\frac{u(s)}{v(s)}\), the quotient rule states \(f'(s) = \frac{v(s)u'(s) - u(s)v'(s)}{(v(s))^2}\). Here, \(u(s) = 4 - 2s^2\) and \(v(s) = 1 - s\).

Step 3: Differentiate the Numerator

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

Step 4: Differentiate the Denominator

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

Step 5: Substitute in the Quotient Rule Formula

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

Step 6: Simplify the Expression

Expand both terms in the numerator: \(-4s + 4s^2 + 4 - 2s^2\). Then combine like terms to get \(2s^2 - 4s + 4\). The denominator remains \((1-s)^2\).

Step 7: Final Simplified Expression

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

Key Concepts

Quotient Rule

The quotient rule is a powerful tool in calculus to find the derivative of a function that is the division of two other functions. If you have a function of the form \( f(x) = \frac{u(x)}{v(x)} \), where both \( u(x) \) and \( v(x) \) are differentiable functions, the derivative of \( f(x) \) is given by:

• \( f'(x) = \frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2} \)

This rule allows us to manage the complexity that arises from dividing functions by considering how both the top and bottom contribute to changes in the function.
This formula helps us understand not only how the numerator and denominator change, but also how they interact.
It provides a systematic way to find the derivative of functions presented as quotients.

Rational Function

Rational functions are expressions that involve the division of two polynomials. They are crucial in understanding a variety of real-world situations where such ratios naturally occur, like speed which is distance over time.
The general form of a rational function is \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials.
It's essential that in a rational function, the denominator \( Q(x) \) is not equal to zero.

• If the denominator equals zero, the function becomes undefined there, so we must be careful about the values of \( x \) we handle.
  This characteristic creates vertical asymptotes, places where the function shoots off to positive or negative infinity.

In our exercise above, the function \( f(s) = \frac{4 - 2s^2}{1 - s} \) is an example of a rational function.
The denominator \( 1-s \) must not be zero, so \( s eq 1 \). Understanding rational functions helps in visualizing how the output graph behaves based on the polynomial forms in both the numerator and denominator.

Derivative Computation

Computing derivatives efficiently is crucial in calculus. Derivatives tell us how much a function is changing at any given point, providing insight into the function's behavior and guiding principles in fields like physics and economics.
To compute the derivative of a rational function like \( f(s) = \frac{4 - 2s^2}{1 - s} \):

• First, identify the numerator and denominator as separate functions \( u(s) = 4 - 2s^2 \) and \( v(s) = 1 - s \).
• Differentiate these simple polynomial parts separately: \( u'(s) = -4s \) and \( v'(s) = -1 \).
• Apply the quotient rule, substituting these derivatives into the formula: \( f'(s) = \frac{(1-s)(-4s) - (4-2s^2)(-1)}{(1-s)^2} \).
• Simplify the expression by expanding and combining like terms to reach a final result: \( f'(s) = \frac{2s^2 - 4s + 4}{(1-s)^2} \).

Through these steps, we capture the function's rate of change, important for analyzing patterns and behaviors in dynamic systems.