Problem 31 - Calculus for Biology and Medicine

Question

Differentiate $$ f(N)=(b-1) N^{4}-\frac{N^{2}}{b} $$

Step-by-Step Solution

Answer

The derivative is \( f'(N) = 4(b-1)N^3 - \frac{2N}{b} \).

Step 1: Identify the Function

The given function is \( f(N) = (b - 1) N^4 - \frac{N^2}{b} \). This function consists of a polynomial with two terms.

Step 2: Differentiate the First Term

The first term is \((b-1) N^4\). To differentiate, apply the power rule: \( \frac{d}{dN}(N^n) = nN^{n-1}\). Thus, \( \frac{d}{dN}((b-1)N^4) = 4(b-1)N^3 \).

Step 3: Differentiate the Second Term

The second term is \(-\frac{N^2}{b}\). Applying the power rule here gives \( \frac{d}{dN}\left(-\frac{N^2}{b} \right) = -2 \frac{N}{b}\).

Step 4: Combine the Derivatives

Now, combine the derivatives of the first and second terms to find \( f'(N)\). The result is: \[ f'(N) = 4(b-1)N^3 - \frac{2N}{b} \].

Key Concepts

Power Rule

The power rule is a fundamental technique in differentiation that simplifies the process of finding the derivative of terms where the variable is raised to a power. It's a simple yet powerful rule to remember. The power rule states:

• If you have a term in the form of \(N^n\), its derivative is \(nN^{n-1}\).

In this exercise, we applied the power rule to differentiate each term of the polynomial function separately.
This straightforward technique allows you to easily handle terms without complicated calculus. For instance, if you consider the first term \((b - 1) N^4\), applying the power rule gives us \(4(b - 1)N^3\). Similarly, for the second term, \(-\frac{N^2}{b}\), the derivative becomes \(-2 \frac{N}{b}\).
By mastering the power rule, differentiation becomes a breeze, especially when dealing with polynomials or any term with exponentiation.

Polynomial Differentiation

Polynomial differentiation is a method focused on finding the derivative of polynomial expressions. Polynomials are algebraic expressions consisting of variables raised to positive integer exponents, and they can have multiple terms combined together. The process involves using rules like the power rule repeatedly.
In our given exercise, the function \(f(N) = (b-1) N^4 - \frac{N^2}{b}\) is a polynomial consisting of a couple of terms. The differentiation process entails handling each term individually. By following this approach, each term's derivative is computed separately:

• The derivative of \((b-1)N^4\) is computed using the power rule as \(4(b-1)N^3\).
• Similarly, the derivative of \(-\frac{N^2}{b}\) is \(-2 \frac{N}{b}\).

This methodical approach ensures that polynomials, no matter the number of terms, are differentiated smoothly, ensuring clarity and correctness in problem-solving.

Derivative of a Polynomial

The derivative of a polynomial is a function that represents the rate of change of the original polynomial function. Understanding the derivative helps you explore how small changes in the input (here, \(N\)) influence changes in the output of the function. This concept is pivotal in calculus and analysis, offering insights into the behavior and characteristics of functions.
For polynomials, finding the derivative involves breaking down the expression into its individual terms, each addressed with differentiation techniques. In our case:

• For \((b-1) N^4\), the derivative is found by applying the power rule, resulting in \(4(b-1)N^3\).
• For \(-\frac{N^2}{b}\), its derivative becomes \(-2 \frac{N}{b}\).

Combining these results, we get the full derivative: \( f'(N) = 4(b-1)N^3 - \frac{2N}{b} \). This expression can now be used to analyze how the original function varies at different points, providing valuable information about trends like increasing or decreasing behaviors.