Problem 3 - The Calculus with Analytic Geometry

Question

In Exercises 1 through 10, find the first and second derivative of the function defined by the given equation. $$ g(s)=2 s^{4}-4 s^{3}+7 s-1 $$

Step-by-Step Solution

Answer

The first derivative is \( g'(s) = 8s^3 - 12s^2 + 7 \), and the second derivative is \( g''(s) = 24s^2 - 24s \).

Step 1 - Identify the function

The given function is: \[ g(s) = 2s^4 - 4s^3 + 7s - 1 \]

Step 2 - Differentiate using power rule (first derivative)

Apply the power rule for differentiation, which states that if \( f(x) = ax^n \), then \( f'(x) = anx^{n-1} \): \[ g(s) = 2s^4 - 4s^3 + 7s - 1 \] Differentiate each term: \[ g'(s) = \frac{d}{ds}(2s^4) - \frac{d}{ds}(4s^3) + \frac{d}{ds}(7s) - \frac{d}{ds}(1) \] Therefore: \[ g'(s) = 8s^3 - 12s^2 + 7 \]

Step 3 - Differentiate again using power rule (second derivative)

Differentiate the first derivative to find the second derivative: \[ g'(s) = 8s^3 - 12s^2 + 7 \] Apply the power rule to each term: \[ g''(s) = \frac{d}{ds}(8s^3) - \frac{d}{ds}(12s^2) + \frac{d}{ds}(7) \] Therefore: \[ g''(s) = 24s^2 - 24s \]

Key Concepts

First Derivative

The first derivative of a function measures the rate at which the function's value changes as the input changes. It's like asking, 'How fast is this changing?' For the function given in the exercise, \( g(s) = 2s^4 - 4s^3 + 7s - 1 \), we want to find \( g'(s) \). To do this, we use calculus differentiation rules, specifically the power rule.
The power rule states that if you have a term in the form of \( ax^n \), its derivative is \( anx^{n-1} \). Let's apply this to each term of the function:

• For \( 2s^4 \): The derivative is \( 4 \times 2s^{4-1} = 8s^3 \).
• For \( -4s^3 \): The derivative is \( 3 \times -4s^{3-1} = -12s^2 \).
• For \( 7s \): The derivative is simply 7.
• For the constant term \( -1 \): The derivative is 0 since constants do not change.

Combining these, we have the first derivative: \( g'(s) = 8s^3 - 12s^2 + 7 \).

Second Derivative

The second derivative gives us information about the curvature of the function or how the rate of change itself is changing. For instance, it can tell us about the concavity and inflection points of the function. To find the second derivative of the given function, we take the derivative of the first derivative we found earlier \( g'(s) = 8s^3 - 12s^2 + 7 \).
Using the power rule again:

• For \( 8s^3 \): The derivative is \( 3 \times 8s^{3-1} = 24s^2 \).
• For \( -12s^2 \): The derivative is \( 2 \times -12s^{2-1} = -24s \).
• For the constant term 7: The derivative is 0.

Combining these terms, we find the second derivative: \( g''(s) = 24s^2 - 24s \).

Power Rule

The power rule is one of the simplest yet most useful rules in differentiation. If you have a term in the form \( ax^n \), the derivative is calculated as \( anx^{n-1} \). This rule applies to any power function, making it crucial for differentiating polynomial expressions.
Here's a step-by-step guide on how to apply the power rule:

• Identify the coefficient \( a \) and the exponent \( n \) from the term.
• Multiply the coefficient \( a \) by the exponent \( n \) to get the new coefficient.
• Subtract one from the exponent \( n \) to get the new exponent.

Here's an example using the term \( 2s^4 \) from our function \( g(s) \):

• Identify \( a = 2 \text{ and } n = 4 \).
• Multiply: \( 4 \times 2 = 8 \).
• Subtract one from the exponent: \( 4 - 1 = 3 \).
• So the derivative of \( 2s^4 \) is \( 8s^3 \).

This process is applied to each term in a function to find its derivative.