Problem 28 - Calculus for Biology and Medicine

Question

Differentiate $$ f(r)=r s^{2}-r $$ with respect to \(r\). Assume that \(s\) is a constant.

Step-by-Step Solution

Answer

The derivative is \( s^2 - 1 \).

Step 1: Identify the Function and Variables

We are given the function \( f(r) = r s^2 - r \). We need to differentiate this function with respect to \( r \). Here, \( s \) is treated as a constant, and \( r \) is the variable with respect to which we differentiate.

Step 2: Differentiate Each Term Respectfully

Differentiate each term of the function separately with respect to \( r \). The function is \( f(r) = r s^2 - r \). Calculate the derivative of each term.\[ \frac{d}{dr}[r s^2] = s^2 \frac{d}{dr}[r] = s^2 \]and\[ \frac{d}{dr}[-r] = -1 \]

Step 3: Combine the Derivatives

Combine the derivatives obtained from each term. Add the derivative of \( r s^2 \) and \( -r \) together to get the overall derivative of the function.\[ \frac{df}{dr} = s^2 - 1 \]

Key Concepts

Calculus

Calculus is an important branch of mathematics that deals with change. It is essential for studying how different variables affect each other and how certain quantities change over time.
Two primary components of calculus are differentiation and integration. Differentiation is the process of finding the derivative of a function. It helps in determining how a function's output changes in response to changes in its input. On the other hand, integration deals with the accumulation of quantities and finding areas under curves.
Calculus is used in various fields such as physics, engineering, economics, and biology. In these areas, it helps in optimizing solutions, modeling changes, and understanding dynamic systems.

• Differentiation helps measure rates of change like velocity or growth rate.
• Integration often finds total amounts, like total distance or area.

By using calculus, we can explore and predict behavior in complex systems. Understanding these core concepts gives one the tools to solve real-world problems.

Derivative

A derivative is a mathematical concept that measures how a function changes as its input changes. In other words, it gives the rate at which a quantity is changing at any given point.
The derivative of a function is denoted as \(f'(x)\), \(\frac{df}{dx}\), or \(\frac{d}{dx}[f(x)]\). They all mean the same thing: the derivative of function \(f\) with respect to variable \(x\).
For instance, if you're moving a car and want to know your speed at a particular moment, you use the derivative of your position function. This tells you how fast you're moving right then.

• If \(f(x) = x^2\), its derivative \(f'(x) = 2x\) shows how the function increases as \(x\) increases.
• The derivative tells us about slope and tangents to curves on a graph.

The derivative is fundamental in calculus because it offers us crucial information about functions and their rates of change.

Constant Function Differentiation

Differentiating a constant function is straightforward. A constant function is one where the output value doesn't change, no matter how the input changes.
For any constant \(c\), the derivative \(\frac{d}{dx}[c] = 0\). This is because the slope of a constant function’s graph is a flat horizontal line, indicating no change. Since derivatives measure change, the absence of change results in a derivative of zero.
Consider the function \(f(r) = r s^2 - r\) from the original exercise. Here, \(s^2\) is a constant term when differentiating with respect to \(r\).

• Differentiate \(r s^2\): The constant \(s^2\) stays while \(r\) differentiates to 1, so the result is \(s^2\).
• Differentiate \(-r\): This straightforwardly becomes -1 because \(r\) differentiates to 1 and the negative sign remains.

In combining these, you find the overall derivative accounting for both the constant and variable parts. Understanding this concept helps ease calculus problems involving constants.