Problem 28 - Brief Applied Calculus

Question

Find the second derivative of each function. $$ \left(x^{3}+x-1\right)\left(x^{3}+1\right) $$

Step-by-Step Solution

Answer

The second derivative is \( f''(x) = 30x^4 + 12x^2.\)

Step 1: Expand the Function

First, expand the given function using distributive property (FOIL method for binomials). This involves multiplying each term in the first binomial by each term in the second binomial:\[ (x^3 + x - 1)(x^3 + 1) = x^3 \cdot x^3 + x^3 \cdot 1 + x \cdot x^3 + x \cdot 1 - 1 \cdot x^3 - 1 \cdot 1 \]which simplifies to:\[ x^6 + x^3 + x^4 + x - x^3 - 1 = x^6 + x^4 + x - 1. \]

Step 2: Find the First Derivative

Differentiate the expanded function with respect to \(x\). Differentiate each term using the power rule \(\frac{d}{dx}[x^n] = nx^{n-1}\):\[ f(x) = x^6 + x^4 + x - 1 \]\[ f'(x) = 6x^5 + 4x^3 + 1. \]

Step 3: Find the Second Derivative

Differentiate the first derivative to obtain the second derivative using the power rule again:\[ f'(x) = 6x^5 + 4x^3 + 1 \]\[ f''(x) = 30x^4 + 12x^2. \]

Key Concepts

Second Derivative

The second derivative, denoted as \( f''(x) \), involves differentiating a function twice with respect to the variable, in this case, \(x\). This operation is key in analyzing the concavity of functions and determining the nature of critical points for applications like graphing.

• First, find the first derivative \( f'(x) \) using differentiation rules such as the power rule.
• Then, differentiate \( f'(x) \) to find the second derivative \( f''(x) \).

For the given problem, we started with \( f(x) = x^6 + x^4 + x - 1 \). The first derivative was calculated as \( f'(x) = 6x^5 + 4x^3 + 1 \). The second differentiation results in \( f''(x) = 30x^4 + 12x^2 \). This second derivative tells us how the rate of change of the original function is itself changing. If \( f''(x) > 0 \), the function is concave upward, and if \( f''(x) < 0 \), it's concave downward.

Power Rule

The power rule is a basic derivation shortcut in calculus that provides a straightforward method to differentiate functions whose terms are in the form of \(x^n\). This rule states that for any term \(x^n\), the derivative \( \frac{d}{dx}[x^n] \) is \( nx^{n-1} \).

• To apply the power rule, multiply the exponent \(n\) by the coefficient of \(x^n\).
• Subtract one from the exponent \(n\) to get the new exponent.

Let's apply the power rule to \( x^6 \), which yields \( 6x^5 \) as its derivative. It was similarly applied to \( x^4 \), resulting in \( 4x^3 \). This simple but powerful rule allowed for easy computation of both the first and second derivatives in the exercise.

Polynomial Functions

Polynomial functions are expressions consisting of variables raised to whole number powers and coefficients. They include simple monomials, like \( x \), and more complex polynomials, such as \( x^6 + x^4 + x - 1 \). These functions are widely used because they can model a variety of phenomena and are easy to differentiate.

• The degrees of the polynomial terms give insights into the behavior and shape of their graphs.
• Differentiation of polynomials applies rules, such as the power rule, to each term individually, making it straightforward.

In our example, expanding the given function using the distributive property was the first step. The multiplication of polynomials led to \( x^6 + x^4 + x - 1 \), a sorted polynomial form that we then differentiated term by term. This procedure highlights how methods like FOIL help break down more complex operations into manageable parts, making calculus problems simpler to handle.