Problem 31 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find \(D_{x} y\) using the rules of this section. $$ y=\left(5 x^{2}-7\right)\left(3 x^{2}-2 x+1\right) $$

Step-by-Step Solution

Answer

The derivative is \( D_x y = 60x^3 - 30x^2 - 32x + 14 \).

Step 1: Identify the Product Rule

The function you are given, \( y = (5x^2 - 7)(3x^2 - 2x + 1) \), is a product of two functions: \( u = 5x^2 - 7 \) and \( v = 3x^2 - 2x + 1 \). The Product Rule states that the derivative of a product \( uv \) is \( u'v + uv' \).

Step 2: Differentiate Each Function

Differentiate \( u = 5x^2 - 7 \) and \( v = 3x^2 - 2x + 1 \) separately. First, the derivative of \( u \) with respect to \( x \) is \( u' = 10x \). Second, differentiate \( v \) to obtain \( v' = 6x - 2 \).

Step 3: Apply the Product Rule

Now, substitute \( u' = 10x \), \( u = 5x^2 - 7 \), \( v' = 6x - 2 \), and \( v = 3x^2 - 2x + 1 \) into the product rule formula: \( D_x y = (10x)(3x^2 - 2x + 1) + (5x^2 - 7)(6x - 2) \).

Step 4: Simplify Each Term

Expand each term: - First term: \( (10x)(3x^2 - 2x + 1) = 30x^3 - 20x^2 + 10x \).- Second term: \( (5x^2 - 7)(6x - 2) = 30x^3 - 10x^2 - 42x + 14 \).

Step 5: Combine and Simplify

Add the two expanded terms from Step 4: \( 30x^3 - 20x^2 + 10x \) and \( 30x^3 - 10x^2 - 42x + 14 \). Combine like terms to simplify:\( 30x^3 + 30x^3 - 20x^2 - 10x^2 + 10x - 42x + 14 = 60x^3 - 30x^2 - 32x + 14 \).

Key Concepts

Product Rule

The Product Rule is an essential tool in calculus for finding the derivative of functions that are the product of two or more functions. It's like a secret recipe that helps you differentiate products without expanding them completely first. When you have a function that looks like this:

• \( y = u(x) \cdot v(x) \)

The Product Rule tells us to apply the following formula:

• \( D_x y = u'(x) \cdot v(x) + u(x) \cdot v'(x) \)

This means you differentiate one of the functions while keeping the other constant, then swap and do it again, and lastly add those results. The Product Rule is your go-to solution whenever you're dealing with multiplicative combinations, making it indispensable in differentiation tasks.

Derivative

A derivative represents the rate at which a function is changing at any given point and is one of the central ideas of calculus. If you think of graphs, the derivative is the slope of the tangent line to the curve at a particular point. This concept lets us understand how a function behaves and predicts things like speed or growth.In mathematical terms, if you have a function \( f(x) \), its derivative is denoted as \( f'(x) \) or \( \frac{df}{dx} \). The process of finding the derivative is called differentiation. Differentiation provides valuable insights, such as:

• Finding maximum or minimum values of functions
• Understanding how points are connected and how steep they are
• Calculating acceleration or any other rate of change

Thus, derivatives are powerful tools with wide applications in various fields like physics, engineering, and economics.

Polynomial Functions

Polynomial functions are mathematical expressions involving variables raised to non-negative integer powers. They take a form like \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \), where \( a_n, a_{n-1}, \ldots, a_0 \) are constants and \( n \) is a non-negative integer.These functions are straightforward to work with in calculus because they obey familiar rules, making them perfect candidates for differentiation. The degree of the polynomial (the highest power of \( x \)) informs us of the behavior and shape of their graphs:

• Linear (straight line), when the highest degree is 1
• Quadratic (parabolic curve), when the highest degree is 2
• Cubic (S-shaped curve), when the highest degree is 3, and so on

By understanding polynomials, one can predict how they act, how to solve them, and how to transform them with ease.

Differentiation Techniques

Differentiation techniques are the methods used to find derivatives of functions. Some basic rules guide this process, and mastering these can make the journey through calculus smoother. Major techniques include:

• Power Rule: If \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \).
• Constant Rule: The derivative of a constant is 0.
• Sum and Difference Rules: The derivative of the sum/difference of functions is the sum/difference of their derivatives.
• Product Rule: As we've already discussed, it's for differentiating products of functions.
• Quotient Rule: For quotients, where you have: \( \frac{u}{v} \), use: \( \frac{v \cdot u' - u \cdot v'}{v^2} \).

Each of these techniques provides a shortcut to quickly and accurately find the derivative, whether you're working with polynomials, exponentials, trigonometric functions, or any combination therein. Knowing which technique to apply, and when, is key to tackling a wide range of calculus problems.