Problem 8 - Calculus for Biology and Medicine

Question

Differentiate the functions given with respect to the independent variable. $$ g(s)=3-4 s^{2}-4 s^{3} $$

Step-by-Step Solution

Answer

The derivative is \( g'(s) = -8s - 12s^2 \).

Step 1: Understand the Function

The function to differentiate is \( g(s) = 3 - 4s^2 - 4s^3 \). We need to find the derivative with respect to \( s \).

Step 2: Differentiate Constant Terms

The derivative of a constant is zero. Therefore, the derivative of \( 3 \) in the function is 0.

Step 3: Apply Power Rule to Each Term

The Power Rule states that \( \frac{d}{ds}[s^n] = ns^{n-1} \). We apply this rule to \(-4s^2\) and \(-4s^3\).

Step 4: Differentiate \(-4s^2\)

Using the Power Rule: \( \frac{d}{ds}[-4s^2] = -4 \cdot 2 \cdot s^{2-1} = -8s \).

Step 5: Differentiate \(-4s^3\)

Using the Power Rule: \( \frac{d}{ds}[-4s^3] = -4 \cdot 3 \cdot s^{3-1} = -12s^2 \).

Step 6: Combine the Results

Combine the results from the derivatives: 0 (from constant) + (-8s) + (-12s^2) = \(-8s - 12s^2\).

Step 7: Write the Final Derivative Expression

The derivative of \( g(s) = 3 - 4s^2 - 4s^3 \) with respect to \( s \) is \( g'(s) = -8s - 12s^2 \).

Key Concepts

Power Rule in Calculus

When it comes to calculus differentiation, one of the most fundamental rules you'll encounter is the power rule. This rule is essential for finding the derivative of polynomial functions. Using the power rule is straightforward: if you have a term in the form of \( s^n \), the derivative can be found by multiplying the exponent \( n \) by the coefficient and then reducing the exponent by one.
For example:

• For \( s^2 \), the derivative is \( 2s^{2-1} = 2s \).
• For \( s^3 \), the derivative would be \( 3s^{3-1} = 3s^2 \).

This rule simplifies the process of differentiation, especially when dealing with polynomials consisting of several terms. Remember, the key is to apply the power rule term by term until each part of the function has been differentiated.

Derivative of Polynomial Functions

Polynomials are a type of function made up of terms where each term is a constant multiplied by a variable raised to a power. When differentiating polynomial functions, it's important to handle each term separately. This makes polynomial differentiation straightforward, especially when using the power rule.
To differentiate a polynomial:

• Identify each term in the polynomial.
• Apply the power rule to each term individually.
• Combine the differentiated terms to find the final result.

For a function like \( g(s) = 3 - 4s^2 - 4s^3 \), you would break it down as:

• The constant \( 3 \) becomes \( 0 \) because the derivative of any constant is zero.
• Applying the power rule to \(-4s^2\) results in \( -8s \).
• Applying the power rule to \(-4s^3\) results in \( -12s^2 \).

The derivative of the entire function is then obtained by adding these results together.

Step-by-Step Solution for Differentiation

A structured step-by-step solution is crucial for understanding how to differentiate any given polynomial function. Let's break down the exercise of differentiating \( g(s) = 3 - 4s^2 - 4s^3 \) to see how it's done.
Start by identifying and understanding the function and the variable with respect to which you are differentiating. Here, it is the function \( g(s) \) with respect to \( s \). Move on to differentiate each part:

• The constant \( 3 \) becomes \( 0 \) as its derivative is zero.
• Differentiate \( -4s^2 \) using the power rule: multiply \( -4 \) by the exponent \( 2 \) and reduce the exponent by one, resulting in \( -8s \).
• Do the same for \( -4s^3 \), resulting in \( -12s^2 \).

Lastly, combine all the differentiated parts: add \( 0 \), \( -8s \), and \( -12s^2 \) to get \( g'(s) = -8s - 12s^2 \). Following each step carefully ensures a clear understanding of the differentiation process and helps reinforce the use of the power rule effectively.