Problem 31 - Calculus for Biology and Medicine

Question

Differentiate $$ f(N)=(b-1) N^{4}-\frac{N^{2}}{b} $$ with respect to \(N\). Assume that \(b\) is a nonzero constant.

Step-by-Step Solution

Answer

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

Step 1: Apply the Power Rule to Differentiate the First Term

The function is given as \( f(N) = (b-1) N^{4} - \frac{N^{2}}{b} \). To differentiate the first term, \((b-1)N^4\), use the power rule: \((x^n)' = nx^{n-1}\). Thus, \(\frac{d}{dN}((b-1)N^4) = 4(b-1)N^3\).

Step 2: Apply the Power Rule to Differentiate the Second Term

The second term is \( -\frac{N^2}{b} \). Again, apply the power rule: \((N^2)' = 2N\). Given the constant factor \(-\frac{1}{b}\), differentiate as \( \frac{d}{dN}\left(-\frac{1}{b}N^2\right) = -\frac{1}{b} \times 2N = -\frac{2N}{b} \).

Step 3: Combine the Results

To find the derivative of the entire function \( f(N) = (b-1) N^{4}-\frac{N^{2}}{b} \), combine the derivatives from Steps 1 and 2. Thus, \( f'(N) = 4(b-1)N^3 - \frac{2N}{b} \).

Key Concepts

Power Rule

When dealing with the differentiation of functions involving exponents, the Power Rule is a essential tool. It's a fundamental concept in calculus that simplifies the process of differentiation. The Power Rule states that for any function of the form \(f(x) = x^n\), the derivative of this function with respect to \(x\) is \(f'(x) = nx^{n-1}\). Breaking it down:

• The exponent \(n\) becomes the coefficient in front of the new term.
• The exponent is then reduced by one to form the new power of \(x\).

Let's see this in action with our given function. For the term \((b-1)N^4\), we use the Power Rule by bringing down the exponent 4 in front of \(N\) and subtracting one from the exponent. Thus, we get \(4(b-1)N^3\). Similarly, differentiating the term \(-\frac{N^2}{b}\) gives us \(-\frac{2N}{b}\), as explained in the next section.

Constant Rule

In calculus, the Constant Rule is crucial when differentiating terms that involve constant multipliers. The rule states that the derivative of a constant multiplied by a function is simply the constant multiplied by the derivative of that function. To clarify:

• When differentiating a term like \(c \cdot g(x)\), where \(c\) is a constant, the result is \(c \cdot g'(x)\).

For example, in the term \(-\frac{N^2}{b}\), considering \(-\frac{1}{b}\) as a constant, the differentiation involves applying the Power Rule on \(N^2\), which gives us \(2N\), and then multiplying by this constant. Hence, the derivative becomes \(-\frac{2N}{b}\). This operation effectively combines both the constant and power rules to obtain the derivative.

Calculus Steps

In differentiation problems, it's essential to follow systematic steps to ensure accurate results. Calculus steps involve:

• Identifying Terms: Start by breaking down the expression into separate terms, each of which will be differentiated individually.
• Applying Rules: Use appropriate differentiation rules like the Power Rule or Constant Rule for each term separately.
• Combining Results: After finding derivatives for each term, combine them to construct the derivative of the entire function.

In the original exercise, after applying the Power Rule to each term:- We derived \(4(b-1)N^3\) for the first term.- We derived \(-\frac{2N}{b}\) for the second term.Finally, these results are combined to achieve the derivative \(f'(N) = 4(b-1)N^3 - \frac{2N}{b}\). By understanding these calculus steps, students can approach differentiation tasks methodically and with greater confidence.