Problem 14 - Calculus Single Variable

Question

Use the Product Rule to compute the derivative of the given expression with respect to \(x\). (In each of Exercises 7,8,14,16, and 18, do not avoid using the Product Rule by first expanding the product.) $$ \left(x^{2}-5\right)(3 x+2) $$

Step-by-Step Solution

Answer

The derivative is \( 9x^2 + 4x - 15 \).

Step 1: Identify functions for the Product Rule

The Product Rule states that the derivative of a product of two functions is given by the derivative of the first function times the second function plus the first function times the derivative of the second function. In this case, identify the functions as follows: - Let \( u(x) = x^2 - 5 \)- Let \( v(x) = 3x + 2 \).

Step 2: Compute the derivative of u(x)

Calculate the derivative of \( u(x) = x^2 - 5 \). The derivative is \[ u'(x) = \frac{d}{dx} (x^2) - \frac{d}{dx} (5) = 2x - 0 = 2x \].

Step 3: Compute the derivative of v(x)

Calculate the derivative of \( v(x) = 3x + 2 \). The derivative is \[ v'(x) = \frac{d}{dx} (3x) + \frac{d}{dx} (2) = 3 + 0 = 3 \].

Step 4: Apply the Product Rule

Use the Product Rule: \( (uv)' = u'v + uv' \). Substitute the derivatives and functions:\[ (x^2 - 5)'(3x + 2) + (x^2 - 5)(3x + 2)' \]. This becomes:\[ 2x(3x + 2) + (x^2 - 5)(3) \].

Step 5: Simplify the expression

Simplify the expression obtained in Step 4:\[ 2x(3x + 2) = 6x^2 + 4x \]\[ (x^2 - 5)(3) = 3x^2 - 15 \]Combine these results:\[ 6x^2 + 4x + 3x^2 - 15 \].

Step 6: Combine like terms

Combine the terms obtained from simplification:\[ 6x^2 + 3x^2 + 4x - 15 = 9x^2 + 4x - 15 \].

Key Concepts

Derivative Calculation

Calculating derivatives forms the core of differential calculus. It involves finding how a function changes as its input changes. The derivative tells us the rate of change or the slope of the function at any point. In our exercise, the goal is to compute the derivative of the expression \((x^2 - 5)(3x + 2)\). A useful technique in such scenarios is the "Product Rule" in calculus.

• The Product Rule is applied when differentiating a product of two functions. It states: if you have two functions \(u(x)\) and \(v(x)\), their derivative \((uv)'\) is given by: \(u'v + uv'\).
• Here, \(u(x) = x^2 - 5\) and \(v(x) = 3x + 2\).
• We need to find the derivatives \(u'(x)\) and \(v'(x)\), then plug these into the Product Rule formula.

This step sets up for finding how the whole expression changes with changes in \(x\).

Function Identification

Identifying the correct functions within an expression is crucial before applying the Product Rule. This involves decomposing the original expression into two parts, each considered as a function of \(x\).
In our case, we break down the expression \((x^2 - 5)(3x + 2)\) into:

• \(u(x) = x^2 - 5\)
• \(v(x) = 3x + 2\)

Once the functions are identified, the next step is calculating their derivatives. Understanding this decomposition helps streamline further calculations and reduces the chances of errors when applying the derivative rules.

Simplification of Expressions

Simplifying expressions after applying the Product Rule is essential to reach a concise answer. Once the derivative is calculated using the Product Rule, the expression can appear cluttered or extended. Simplification tidies this up, making it easier to comprehend and present.Let's take the expression we have:

• First calculate \(2x(3x + 2)\), resulting in \(6x^2 + 4x\).
• Then calculate \((x^2 - 5)(3)\), resulting in \(3x^2 - 15\).

The simplification involves distributing and organizing terms derived from the application of derivatives, effectively cleaning up the expression for the final step.

Combining Like Terms

Combining like terms is the final touch in evaluating derivatives. After applying the Product Rule and simplifying the expressions involved, you will have an assortment of terms. Grouping these similar terms results in a much simpler expression.In our exercise, the expression obtained was:

• \(6x^2 + 4x + 3x^2 - 15\).
• Combine the \(x^2\) terms: \(6x^2 + 3x^2 = 9x^2\).
• The final expression becomes: \(9x^2 + 4x - 15\).

This step involves adding coefficients of matching terms and prepares our expression for interpretation or further use in problems. It neatly wraps up the calculation and provides the derivative in its simplest form.