Problem 20 - Calculus for Biology and Medicine

Question

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ h(s)=\frac{2(3-s)^{2}}{s^{2}+(7 s-1)^{2}} $$

Step-by-Step Solution

Answer

Differentiate using the quotient rule and simplify.

Step 1: Identify the function

The function given is \( h(s) = \frac{2(3-s)^2}{s^2 + (7s-1)^2} \). It is a quotient of two functions. Let's identify the numerator \( N(s) = 2(3-s)^2 \) and the denominator \( D(s) = s^2 + (7s-1)^2 \).

Step 2: Apply the Quotient Rule

The derivative of a quotient \( \frac{N(s)}{D(s)} \) is given by the quotient rule: \[ h'(s) = \frac{N'(s)D(s) - N(s)D'(s)}{D(s)^2} \]. In our case, we'll need to find \( N'(s) \) and \( D'(s) \).

Step 3: Differentiate the numerator

Calculate \( N'(s) = 2 \cdot [2(3-s) \cdot (-1)] = -4(3-s) = -4 \cdot 3 + 4s = 4s - 12 \).

Step 4: Differentiate the denominator

First expand \( (7s-1)^2 = 49s^2 - 14s + 1 \), then \( D(s) = s^2 + (49s^2 - 14s + 1) = 50s^2 - 14s + 1 \). Now \( D'(s) = 100s - 14 \).

Step 5: Substitute derivatives into the Quotient Rule

Substitute \( N'(s) = 4s - 12 \) and \( D'(s) = 100s - 14 \) into the quotient rule formula: \[ h'(s) = \frac{(4s - 12)(50s^2 - 14s + 1) - 2(3-s)^2(100s - 14)}{(s^2 + (7s-1)^2)^2} \].

Step 6: Simplify the expression

Simplify the expression by expanding and combining like terms, which can be complex. For this exercise, we'll stop here as further hand calculation would require longer expansion.

Key Concepts

Quotient Rule

When dealing with differentiation, the quotient rule is a vital tool in calculus for finding the derivative of a function that is the quotient of two other functions. Specifically, this rule is used to differentiate a function, where one function is divided by another. The quotient rule states that if you have a function represented as \( \frac{N(s)}{D(s)} \), then its derivative can be calculated using: \[ h'(s) = \frac{N'(s)D(s) - N(s)D'(s)}{D(s)^2} \] This formula essentially involves:

• Taking the derivative of the numerator \( N(s) \)
• Multiplying it by the denominator \( D(s) \)
• Subtracting the product of the numerator and the derivative of the denominator \( D'(s) \)
• Finally, dividing the whole expression by the square of the denominator \( D(s)^2 \)

Conceptually, it can be seen as a balance, calculating the rate of change of the ratio between two changing quantities.

Derivative Calculation

Calculating derivatives, especially using the quotient rule, involves a series of steps that require meticulous attention. For the function given, one begins by identifying the numerator and the denominator separately. Once identified, you need to calculate both \( N'(s) \), the derivative of the numerator, and \( D'(s) \), the derivative of the denominator. For this exercise, the differentiation process involves:

• Utilizing standard derivative rules: power rule, product rule, and sum rule dependent on the form of the involved terms
• Ensuring the correct calculation of derivative, which may require algebraic manipulation such as expansion

Knowing each step in the process ensures you correctly apply differentiation rules and obtain accurate derivatives, as seen when we calculated: \( N'(s) = 4s - 12 \) and \( D'(s) = 100s - 14 \).

Numerator and Denominator Differentiation

Differentiating the numerator and the denominator separately is crucial before applying the quotient rule. This requires a good understanding of basic differentiation techniques. **Differentiating the Numerator**: In our case, the numerator is expressed as \( N(s) = 2(3-s)^2 \). The differentiation here involves using the power rule and taking note of the chain rule due to the composition within \((3-s)\). Calculating, we found \( N'(s) = 4s - 12 \). **Differentiating the Denominator**: For the denominator defined by \( D(s) = s^2 + (7s-1)^2 \), differentiation requires expansion first through squaring \((7s-1)^2\) and then summing it with \( s^2 \). The expanded form, \( 50s^2 - 14s + 1 \), is then directly differentiated resulting in \( D'(s) = 100s - 14 \). Understanding these processes is pivotal for applying the quotient rule effectively. Differentiating each component correctly ensures that the ultimate derivative you compute is accurate.