Problem 31 - Calculus for Biology and Medicine

Question

Differentiate $$ g(N)=\frac{b N}{k+N} $$ with respect to \(N\). Assume that \(b\) and \(k\) are positive constants.

Step-by-Step Solution

Answer

The derivative is \(g'(N) = \frac{bk}{(k+N)^2}.\)

Step 1: Recognize the Formula type

The given function \(g(N) = \frac{bN}{k+N}\) is a rational function, which suggests we will need to apply the quotient rule for differentiation.

Step 2: Recall the Quotient Rule

The quotient rule states that if you have a function \(\frac{u}{v}\), where both \(u\) and \(v\) are functions of \(N\), then its derivative is given by: \[\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}.\]

Step 3: Identify \(u\) and \(v\) and their Derivatives

Here, we let \(u = bN\) and \(v = k+N\). Their derivatives with respect to \(N\) are \(u' = b\) and \(v' = 1\).

Step 4: Apply the Quotient Rule

Substituting \(u\), \(v\), \(u'\), and \(v'\) into the quotient rule, we get: \[g'(N) = \frac{b(k+N) - bN}{(k+N)^2}.\]

Step 5: Simplify the Expression

Simplify the numerator: \(b(k+N) - bN\) becomes \(bk\), resulting in: \[g'(N) = \frac{bk}{(k+N)^2}.\]

Step 6: Finalize the Derivative

The final expression for the derivative of the given function is: \[g'(N) = \frac{bk}{(k+N)^2}.\]

Key Concepts

Quotient Rule

The quotient rule is an essential tool in calculus, used to differentiate functions that are expressed as one function divided by another. It allows us to find the derivative of a quotient of two functions.To apply the quotient rule, you need two functions, let's call them \( u \) and \( v \), where both are functions of a variable, usually denoted as \( x \) or \( N \). The derivative of the function \( \frac{u}{v} \), denoted as \( \left(\frac{u}{v}\right)' \), is calculated using the formula:\[\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}\]

• \( u' \) is the derivative of the numerator function \( u \).
• \( v' \) is the derivative of the denominator function \( v \).
• \( v^2 \) means you square the denominator function \( v \).

Make sure when using the quotient rule that the denominator \( v \) is never zero, as this can make the expression undefined. Understanding each component will make applying the quotient rule much easier.

Derivative

The derivative of a function measures how the function's output value changes as its input changes. In simpler terms, it's the rate of change or slope of the function. In calculus, finding the derivative is one of the fundamental operations and is crucial for understanding behaviors of different functions.For a function \( g(N) \), the derivative, denoted as \( g'(N) \), provides important information:

• The slope of the tangent to the curve at any given point.
• How exactly the function grows or shrinks as the variable \( N \) changes.

In the context of different functions like polynomials, exponential functions, and rational functions, different rules (product, chain, and quotient) are used to find derivatives effectively. Once you have the derivative, you can analyze critical points, optimize functions, and solve a variety of real-world problems.

Rational Function

A rational function is a type of function represented as the division of two polynomials. In general, if you have a function of the form \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials, then you have a rational function.Rational functions have unique characteristics:

• The numerator and denominator both consist of polynomial expressions of varying degrees.
• The points where the denominator equals zero are called 'poles', which cause the function to be undefined at those points.
• Rational functions can have vertical or horizontal asymptotes based on the relationships between the degrees of the polynomials in the numerator and the denominator.

When dealing with rational functions in calculus, particularly when differentiating, the quotient rule often comes into play. This is because the function's form naturally aligns with the rule's requirements. Understanding how to manipulate and differentiate these functions is key to solving many calculus and real-world problems.