Problem 17 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify Differentiation Rules

To differentiate the function \( f(s) = s^{3} e^{3} + 3e \), we must identify which differentiation rules to apply. Since \( e^{3} \) and \( e \) are constants and \( s^{3} \) is a variable term, this will involve the power rule and the constant rule.

Step 2: Differentiate Each Term Separate

1. Differentiate \( s^3 e^3 \): Recognize that \( e^3 \) is constant, so use the constant multiple rule along with the power rule: \( \frac{d}{ds}[s^3 e^3] = e^3 \cdot \frac{d}{ds}[s^3] = e^3 \cdot 3s^2 = 3s^2 e^3 \).2. Differentiate \( 3e \): The whole term is a constant, hence its derivative is \( 0 \).

Step 3: Combine Results

Combine the derivatives of both terms: the derivative of \( f(s) \) is \( 3s^2 e^3 + 0 = 3s^2 e^3 \). Thus, the derivative of the function is \( f'(s) = 3s^2 e^3 \).

Key Concepts

Power Rule

The power rule is a fundamental tool in calculus for differentiating functions of the form \( x^n \), where \( n \) is any real number. When you have a function \( f(x) = x^n \), the power rule states that its derivative is \( f'(x) = nx^{n-1} \). This rule makes finding the slope of polynomial functions straightforward.

For example, let's consider the term \( s^3 \) from our exercise. According to the power rule, the derivative of \( s^3 \) is calculated as follows:

• Identify the exponent \( n \), which is 3.
• Multiply the term by \( n \): \( 3s^{3-1} \).
• Simplify to get \( 3s^2 \).

This process allows you to easily handle more complex functions that have variable terms raised to a power. Remember, the power rule can only be applied when differentiating terms with variable bases.

Constant Rule

The constant rule in differentiation helps you deal with parts of a function that don’t change based on the variable. Simply put, the derivative of a constant is always zero, since constants do not vary.

In our original function, we have the term \( 3e \), where both 3 and \( e \) are constants. When differentiating such a term, you apply the constant rule which states:

• The derivative of any constant \( c \) is zero.

Thus, when calculating the derivative of \( 3e \), it results in zero because there is no variable to affect the expression as it changes. This rule helps eliminate constant terms from consideration when identifying changes in the function's slope.

Calculus Function

In calculus, functions play a crucial role in understanding change and motion. A calculus function represents a mathematical relationship where each input (or 'independent variable') corresponds to a unique output. Functions are usually expressed in terms like \( f(x) \), where \( x \) is the variable.

In the differentiation process, functions like \( f(s) = s^3 e^3 + 3e \) are analyzed to determine how they change as the variable \( s \) changes. This involves applying various rules of differentiation, such as the power and constant rules, to find the function's derivative. The derivative \( f'(s) \) describes the rate at which the function's output changes with respect to \( s \).

In our case, understanding how to differentiate the given function allows us to see how \( f(s) \) behaves as \( s \) changes. This detailed process shows how calculus functions reveal more than just static values—they provide insight into the behavior and dynamics of mathematical relationships.