Problem 112 - Calculus Volume 1

Question

For the following exercises, find \(f^{\prime}(x)\) for each function. $$f(x)=(x+2)\left(2 x^{2}-3\right)$$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 6x^2 + 8x - 3 \).

Step 1: Identify the Function Type and Formula

The given function is a product of two distinct functions: \( (x+2) \) and \( (2x^2 - 3) \). Thus, we'll use the Product Rule for differentiation, which states: If \( f(x) = g(x)h(x) \), then \( f'(x) = g'(x)h(x) + g(x)h'(x) \).

Step 2: Differentiate the First Function

The first function is \( g(x) = x + 2 \). The derivative of \( g(x) \) is \( g'(x) = 1 \) because the derivative of \( x \) with respect to \( x \) is 1, and the derivative of a constant like 2 is zero.

Step 3: Differentiate the Second Function

The second function is \( h(x) = 2x^2 - 3 \). Using the power rule, the derivative \( h'(x) = 4x \) since the derivative of \( 2x^2 \) is \( 4x \) and the derivative of \(-3\) is zero.

Step 4: Apply the Product Rule Formula

Using the derivatives calculated and substituting into the product rule formula: \[ f'(x) = g'(x)h(x) + g(x)h'(x) \] Substitute \( g'(x) = 1 \), \( h(x) = 2x^2 - 3 \), \( g(x) = x + 2 \), and \( h'(x) = 4x \):\[ f'(x) = 1(2x^2 - 3) + (x+2)(4x) \].

Step 5: Simplify the Expression

Now, simplify the expression obtained from Step 4: \( f'(x) = 2x^2 - 3 + 4x(x + 2) \). Expand \( 4x(x + 2) \) to get \( 4x^2 + 8x \). Thus, \( f'(x) = 2x^2 - 3 + 4x^2 + 8x \). Combine like terms:\[ f'(x) = (2x^2 + 4x^2) + 8x - 3 \] \[ f'(x) = 6x^2 + 8x - 3 \].

Step 6: Final Result

The derivative of the given function, using the product rule and simplification, is: \( f'(x) = 6x^2 + 8x - 3 \). It's advisable to validate the expression by reviewing calculations.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that deals with deriving a new function from an existing one. This process helps in determining the rate of change of a quantity. For the function provided, we have two parts:

• \((x+2)\)
• \((2x^2 - 3)\).

Differentiation provides us with a tool to find how each part contributes to the change in the whole function through the use of derivatives.
In this context, differentiation involves identifying the type of function you’re dealing with (here, a product of two functions) and using appropriate rules (like the product rule) to find the derivative. Differentiating sums and products of functions relies heavily on foundational rules like the power rule and product rule, which guide how each term is treated in the differentiation process.

Power Rule

The power rule is a basic yet powerful tool in calculus for finding the derivative of polynomial functions. It states that for any function of the form \(x^n\), the derivative is \(nx^{n-1}\).
Applying the power rule to \(2x^2 - 3\), we focus on \(2x^2\). Here, \(n\) is 2, so its derivative is calculated as \(4x\) (using the power rule). The constant \(-3\) has no change, as constants have derivatives of zero.
The power rule simplifies the process of differentiation by reducing each term one exponent level, enabling us to handle more complex functions through simplicity in breaking down each term.

Derivative Calculation

Now, let's focus on calculating the derivative of the full function using the product rule. Once you know the derivatives of each individual function part, they are combined using the product rule formula.
The product rule states: if you have a function \(f(x) = g(x)h(x)\), then \(f'(x) = g'(x)h(x) + g(x)h'(x)\).

• First, identify \(g(x) = x + 2\) with its derivative \(g'(x) = 1\).
• Next, for \(h(x) = 2x^2 - 3\), its derivative is \(h'(x) = 4x\).

Combine these using the product rule: substituting into the formula provides you with \(f'(x) = 1(2x^2 - 3) + (x+2)(4x)\).
Finally, simplify the expression \(2x^2 - 3 + 4x(x + 2)\) to yield the smoothest form of the derivative, \(6x^2 + 8x - 3\), showcasing how each rule stacks together to provide a complete solution.