Problem 38 - Calculus for Biology and Medicine

Question

Differentiate $$ g(N)=r N\left(1-\frac{N}{K}\right) $$

Step-by-Step Solution

Answer

The derivative is \( g'(N) = r - \frac{2rN}{K} \).

Step 1: Identify the Derivative Rules

The function is given as \( g(N) = rN(1 - \frac{N}{K}) \). Here, \( r \) and \( K \) are constants. You'll be using the product rule for differentiation, which in general for two functions \( u(N) \) and \( v(N) \) is given by \( (uv)' = u'v + uv' \). Here, let \( u(N) = rN \) and \( v(N) = (1 - \frac{N}{K}) \).

Step 2: Differentiate Using the Product Rule

Differentiate \( u(N) \): \( u(N) = rN \Rightarrow u'(N) = r \). Next, differentiate \( v(N) \): \( v(N) = 1 - \frac{N}{K} \Rightarrow v'(N) = -\frac{1}{K} \). Apply the product rule: \( g'(N) = u'(N)v(N) + u(N)v'(N) = r(1 - \frac{N}{K}) + rN(-\frac{1}{K}) \).

Step 3: Simplify the Derivative

Simplify the expression: \( g'(N) = r(1 - \frac{N}{K}) - \frac{rN}{K} = r - \frac{rN}{K} - \frac{rN}{K} \). Combine terms: \( g'(N) = r - 2\frac{rN}{K} \).

Step 4: Finalize the Derivative Expression

The derivative of the given function is \( g'(N) = r - \frac{2rN}{K} \). This represents the rate of change of the function with respect to \( N \).

Key Concepts

Understanding the Product Rule

In calculus, the product rule is an essential technique used to differentiate functions that are the product of two or more differentiable functions. The rule states that if you have two functions, say \( u(N) \) and \( v(N) \), their derivative can be obtained using the formula:

• \((uv)' = u'v + uv'\)

This means that to find the derivative of a product of two functions, you differentiate the first function and multiply by the second, then add the product of the first function and the derivative of the second.
This strategy is instrumental because many real-world functions are the result of multiplying different factors that vary with the same variable.
The product rule helps break down these complex relationships into simpler parts, making analysis easier.

Exploring Calculus

Calculus is a powerful branch of mathematics that focuses on studying change. It is divided into two main parts:

• **Differential Calculus**: Concerned with finding the rate at which variables change. It deals primarily with derivatives and their applications.
• **Integral Calculus**: Concerned with the accumulation of quantities, such as areas under curves or total accumulated change, using integrals.

Differential calculus focuses on how a function changes as its input changes. For example, it can be used to find how the velocity of an object changes with time, requiring the calculation of derivatives.
Mastering calculus concepts like differentiation is key to understanding and solving real-world problems involving rates of change and dynamic systems.

Mastering Derivative Rules

Derivative rules are fundamental tools in calculus used to compute the derivative at any given point of a function. These rules streamline the differentiation process. Some of the core derivative rules include:

• **Power Rule**: Used for functions raised to a power, \( \frac{d}{dN} [N^n] = nN^{n-1} \).
• **Constant Rule**: If \( c \) is a constant, then \( \frac{d}{dN} [c] = 0 \).
• **Sum Rule**: The derivative of a sum is the sum of the derivatives, \( (f+g)' = f' + g' \).
• **Product Rule**: As seen, applied to products of two functions.
• **Quotient Rule**: Used for functions divided by another, \( \frac{d}{dN} \left[ \frac{f}{g} \right] = \frac{f'g - fg'}{g^2} \).

Learning and applying these rules allows for the efficient computation of derivatives, which leads to a deeper understanding of the behavior and properties of functions.