Problem 17 - Calculus for Biology and Medicine

Question

Differentiate the functions given in Problems with respect to the independent variable. $$ f(s)=s^{3} e^{3}+3 e $$

Step-by-Step Solution

Answer

The derivative is \( 3s^2 e^3 \).

Step 1: Identify the function to differentiate

We are given the function \( f(s) = s^3 e^3 + 3e \). We need to differentiate it with respect to \( s \).

Step 2: Differentiate each term

The derivative of a constant is zero. Here, \( e^3 \) is a constant, so differentiate \( s^3 \cdot e^3 \) using the power rule: \( \frac{d}{ds}(s^3 \cdot e^3) = 3s^2 \cdot e^3 \). The derivative of \( 3e \), which is also a constant, is 0.

Step 3: Combine the derivatives

Combine the derivatives from the previous step. The derivative of \( f(s) = s^3 e^3 + 3e \) is \( 3s^2 e^3 + 0 \).

Key Concepts

Understanding the Power Rule

The power rule is a fundamental concept in calculus used for finding derivatives, especially when dealing with polynomial functions. It makes differentiation quite simple by providing a straightforward formula. Here's how it works: To differentiate a function of the form \(x^n\), where \(n\) is any real number, you multiply by the exponent \(n\), then decrease the power of \(x\) by one. Mathematically, this is shown as: \[ \frac{d}{dx}(x^n) = n \cdot x^{n-1} \]When applying the power rule, it's essential to ensure the function is expressed in terms of a single variable raised to a power. For example, if we want to differentiate \(s^3\), we apply the rule directly:

• Start with the exponent: 3
• Multiply it by the variable: \(3 \cdot s^{3-1}\)
• Simplify to get \(3s^2\)

This technique significantly speeds up the process of finding derivatives, especially for functions with higher powers.

Exploring the Constant Rule

The constant rule in differentiation is a simple yet powerful tool employed when dealing with functions that include constant terms. It states that the derivative of a constant with respect to any variable is zero. In calculus, a constant is a value that does not change. It is independent of the variable of differentiation. For instance, in the function \(f(s) = s^3 e^3 + 3e\), constants are \(e^3\) and \(3e\). The constant rule can be summarized as:\[ \frac{d}{dx}(c) = 0 \]where \(c\) is a constant. Here's a step-by-step breakdown:

• Identify the constant: \(3e\)
• Note its derivative is zero: \(\frac{d}{ds}(3e) = 0\)
• Apply the same to any term solely multiplied by constants: \(e^3\)

This rule simplifies the differentiation of functions with additive constant terms by removing them from the differentiation process straightforwardly.

Derivative Calculation Made Simple

In calculus, derivative calculation is the process of finding the derivative of a function, which represents the rate at which the function's output changes concerning its input. It's a crucial concept in understanding changes and optimizing functions. Let's break down the calculations using both the power rule and constant rule.When differentiating a function, we dissect it term by term. Consider the function \(f(s) = s^3 e^3 + 3e\):- For the term \(s^3 e^3\):\(e^3\) is a constant, so first, treat it as a constant coefficient and focus on differentiating \(s^3\) using the power rule. This gives us \(3s^2 e^3\).
- For the constant term \(3e\), use the constant rule which tells us that the derivative is 0 because it doesn't change as \(s\) changes.After differentiation, we combine the results:- \(f'(s) = 3s^2 e^3 + 0\), which simplifies to \(3s^2 e^3\).This methodical approach ensures we consider each part of the function, applying appropriate rules to extract the derivative effectively.