Problem 32 - Calculus for Biology and Medicine

Question

Differentiate $$ g(N)=\frac{N}{(k+b N)^{2}} $$ with respect to \(N\). Assume that \(b\) and \(k\) are positive constants.

Step-by-Step Solution

Answer

The derivative is \( g'(N) = \frac{k + bN - 2Nb}{(k+bN)^3} \).

Step 1: Identify the Rule to Use

The function to differentiate is \( g(N) = \frac{N}{(k + bN)^2} \). This is a quotient, so we'll use the quotient rule for differentiation. The quotient rule states if \( h(N) = \frac{u(N)}{v(N)} \), then \( h'(N) = \frac{u'(N)v(N) - u(N)v'(N)}{(v(N))^2} \). Here, \( u(N) = N \) and \( v(N) = (k + bN)^2 \).

Step 2: Differentiate the Numerator

Differentiate the numerator function, \( u(N) = N \). The derivative of \( N \) with respect to \( N \) is 1. Thus, \( u'(N) = 1 \).

Step 3: Differentiate the Denominator

Differentiate the denominator function, \( v(N) = (k + bN)^2 \). Use the chain rule. Let \( w(N) = k + bN \), so \( v(N) = w(N)^2 \). Then \( w'(N) = b \) and by the chain rule, \( v'(N) = 2w(N)w'(N) = 2(k + bN)b \).

Step 4: Apply the Quotient Rule

Substitute \( u(N), v(N), u'(N), \) and \( v'(N) \) into the quotient rule formula: \[ g'(N) = \frac{1 \cdot (k + bN)^2 - N \cdot 2(k + bN)b}{((k + bN)^2)^2} \] Simplify the expression to get the derivative of \( g(N) \).

Step 5: Simplify the Expression

Simplify the numerator: \[ 1 \cdot (k + bN)^2 - N \cdot 2(k + bN)b = (k + bN)^2 - 2Nb(k + bN) \] Finally, combine and simplify further if possible to achieve a more compact form. This gives the final simplified form of \( g'(N) \).

Step 6: Final Simplified Form

The derivative is: \[ g'(N) = \frac{(k+bN)^2 - 2Nb(k+bN)}{(k+bN)^4} = \frac{k + bN - 2Nb}{(k+bN)^3} \].

Key Concepts

Quotient Rule

In calculus, the Quotient Rule is a method to differentiate functions that are divided by one another. When you have a function of the form \( h(N) = \frac{u(N)}{v(N)} \), where both \( u(N) \) and \( v(N) \) are differentiable functions, the derivative, \( h'(N) \), can be found using the formula:

• \( h'(N) = \frac{u'(N)v(N) - u(N)v'(N)}{(v(N))^2} \)

This formula might look a bit complicated but here's a simple approach to use it:- Differentiate the numerator function, \( u(N) \), to get \( u'(N) \).- Differentiate the denominator function, \( v(N) \), to get \( v'(N) \).- Substitute these derivatives into the formula.Using this method to differentiate \( g(N) = \frac{N}{(k+bN)^2} \), we identified \( u(N) = N \) and \( v(N) = (k + bN)^2 \). The differentiation steps follow directly from the formula, leading us to apply the quotient rule to get our expression for \( g'(N) \).
Remember, the key point is to carefully determine \( u(N) \) and \( v(N) \), and ensure they're correctly differentiated.

Chain Rule

The Chain Rule in differentiation is a technique used for finding the derivative of composite functions. A composite function is when one function is nested inside another. For instance, in \( v(N) = (k + bN)^2 \), the inner function is \( w(N) = k + bN \) and it's raised to the power of 2, forming the outer function.The Chain Rule states that to differentiate a composite function \( f(g(N)) \), you find:

• \( (f(g(N)))' = f'(g(N)) \cdot g'(N) \)

For our exercise, to differentiate \( v(N) \), we consider it as \( f(w(N)) = w(N)^2 \), with inner function \( w(N) = k + bN \). This process involves two steps:
- First, differentiate the outer function assuming the inner function is a variable: \( f'(w(N)) = 2w(N) \).- Next, differentiate the inner function: \( w'(N) = b \).Multiplying these two results gives \( v'(N) = 2(k + bN)b \), completing the chain rule application.

When using the Chain Rule, it is vital to identify which part is your inner and which is the outer function. This clarity will make applying the rule straightforward and avoid mistakes.

Derivative Simplification

After successfully applying differentiation rules, the resultant expression often requires simplification to be more understandable or usable. In our exercise, the quotient rule produced an initial derivative that's a complex fraction:

• \( g'(N) = \frac{1 \cdot (k + bN)^2 - N \cdot 2(k + bN)b}{((k+bN)^2)^2} \)

Simplifying derivatives generally involves:- Combining like terms and simplifying by factoring.- Canceling out terms if applicable.- Reducing complex fractions to simpler forms.
Starting with our numerator, \( (k + bN)^2 - 2Nb(k + bN) \), it helps to distribute and combine like terms if possible.
Here, realizing \( 1 \cdot (k + bN)^2 \) simplifies to \( (k + bN)^2 \), which then can be effectively simplified by reducing common terms with \( 2Nb(k + bN) \).This yields the simpler form:

• \( g'(N) = \frac{k + bN - 2Nb}{(k+bN)^3} \)

Effective simplification not only makes the expression easier to interpret, but it also provides clearer insights into how changes in \( N \) affect the value of the derivative.