Problem 13 - Brief Applied Calculus

Question

Find the second derivative of each function. $$ f(x)=\left(x^{2}-2\right)\left(x^{2}+3\right) $$

Step-by-Step Solution

Answer

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

Step 1: Identify the product rule

Since the function is a product of two functions, \( f(x) = (x^2 - 2)(x^2 + 3) \), we need to recognize that we'll apply the product rule to differentiate it. The product rule states that if you have two functions \( u(x) \) and \( v(x) \), then the derivative is \( u'(x)v(x) + u(x)v'(x) \). Here, let \( u(x) = x^2 - 2 \) and \( v(x) = x^2 + 3 \).

Step 2: Differentiate each function independently

Differentiate each function: \( u(x) = x^2 - 2 \) gives \( u'(x) = 2x \), and \( v(x) = x^2 + 3 \) gives \( v'(x) = 2x \).

Step 3: Apply the product rule

Using the product rule, find \( f'(x) \):\[ f'(x) = u'(x)v(x) + u(x)v'(x) = (2x)(x^2 + 3) + (x^2 - 2)(2x) \].

Step 4: Simplify the expression

Expand the expression:\[ f'(x) = 2x(x^2 + 3) + (x^2 - 2)(2x) = 2x^3 + 6x + 2x^3 - 4x \].Combine like terms:\[ f'(x) = 4x^3 + 2x \].

Step 5: Differentiate the first derivative to find the second derivative

Differentiate \( f'(x) = 4x^3 + 2x \) to find the second derivative:\[ f''(x) = rac{d}{dx}(4x^3) + rac{d}{dx}(2x) = 12x^2 + 2 \].

Step 6: Conclusion: State the second derivative

The second derivative of the function \( f(x) = (x^2 - 2)(x^2 + 3) \) is \( f''(x) = 12x^2 + 2 \).

Key Concepts

Product Rule

In calculus, the product rule is a vital tool for differentiating functions that are the product of two or more functions. When you have a function that can be expressed as the product of two functions, like in our exercise where \( f(x) = (x^2 - 2)(x^2 + 3) \), you cannot simply differentiate each factor separately and multiply the results. The product rule says that if you have two functions \( u(x) \) and \( v(x) \), the derivative of the product \( u(x)v(x) \) is given by\[ u'(x)v(x) + u(x)v'(x) \]. This rule ensures that both the rate of change of \( u \) and \( v \) are considered, along with their respective multipliers, producing an accurate derivative of the product.Applying this to our functions,\( u(x) = x^2 - 2 \) and \( v(x) = x^2 + 3 \), helps us find the correct derivative. Recognizing when to apply the product rule is crucial in solving problems involving products of functions.

Differentiation

Differentiation is the process of finding the derivative of a function. A derivative represents the rate at which a quantity changes. Mathematically, it shows us how the function's output value changes as its input changes.The main steps in differentiation involve applying rules such as the power rule, product rule, quotient rule, and chain rule. In our problem, we first establish that the product rule applies, as we're dealing with a product of two functions.To differentiate effectively, each part of the product must be addressed:

• First, take the derivative of \( u(x)=x^2 - 2 \), which gives \( u'(x)=2x \), using the power rule.
• Next, differentiate \( v(x)=x^2+3 \) to get \( v'(x)=2x \).

Following these individual steps makes the process of finding derivatives organized and methodical, easing complexity in more advanced problems.

First Derivative

The first derivative of a function, noted as \( f'(x) \), represents the slope of the tangent line to the curve at any point \( x \). This gives us an idea of how the function is changing at that specific location.For the function given, our task was to find \( f'(x) \) of \( f(x) = (x^2 - 2)(x^2 + 3) \) using the product rule:\[ f'(x) = u'(x)v(x) + u(x)v'(x) \].After substitution, we have:\[ f'(x) = (2x)(x^2 + 3) + (x^2 - 2)(2x) \].Upon simplification:

• Expand to get \( 2x^3 + 6x + 2x^3 - 4x \)
• Combine like terms to find the simplified form: \( f'(x) = 4x^3 + 2x \)

With \( f'(x) \) known, you can determine how the function behaves over its domain, such as increasing, decreasing, or identifying any local maximum or minimum points.