Question

Some of these can be multiplied out. For a few of these, take the derivative both before and after multiplying out, and compare the two. $$y=(7-2 x)(x+4)$$

Short answer

After expanding and simplifying, the function becomes \(y=7x+28-2x^2\). The derivative is \(y'=7-4x\). Using the product rule directly, the derivative is also \(y'=7-4x\). Both methods yield the same derivative.

Step-by-step solution

Expansion

Expand the function using the distributive property (FOIL method for binomials), where each term in the first parenthesis is multiplied by each term in the second parenthesis.

Combine Like Terms

After expansion, combine like terms to simplify the function into standard polynomial form.

Differentiate the Expanded Form

Take the derivative of the expanded polynomial form using the power rule, which states that the derivative of \(x^n\) is \(nx^{n-1}\).

Differentiate using the Product Rule

Instead of expanding, use the product rule to differentiate the original function. The product rule states that the derivative of a product of two functions is the derivative of the first times the second plus the first times the derivative of the second.

Compare Results

Compare the results from Steps 3 and 4 to confirm that they are the same, thereby demonstrating that the derivative of the function is the same whether the function is expanded first or not.

Key concepts

Distributive Property

The distributive property is a cornerstone in algebra and is essential when expanding expressions. It allows you to multiply a single term by each term in a parenthesis, effectively 'distributing' the multiplication across the terms within the parenthesis. For example, if you have an expression like \(a(b + c)\), it simplifies to \(ab + ac\).

When working with polynomials, as seen in the provided exercise \((7 - 2x)(x + 4)\), you use the distributive property to multiply each term in the first parenthesis by each term in the second parenthesis, which results in a new polynomial. Understanding the distributive property is critical for correctly expanding expressions and simplifying them before further operations, such as differentiation, can be performed.

FOIL Method

The FOIL method stands for First, Outer, Inner, Last. It's a handy mnemonic for remembering the order in which to multiply terms when you're dealing with binomials (expressions with two terms). When you have an expression like \((a + b)(c + d)\), the FOIL method directs you to:

• First: Multiply the first terms in each parenthesis: \(ac\).
• Outer: Multiply the outer terms: \(ad\).
• Inner: Multiply the inner terms: \(bc\).
• Last: Multiply the last terms in each parenthesis: \(bd\).

By combining these products, you'll have your expanded polynomial. The FOIL method is essentially an application of the distributive property specific to binomials, making it a useful tool for problems like the original exercise.

Power Rule

The power rule is a fundamental tool in differential calculus for finding derivatives of monomials, where a monomial is a single term like \(x^n\). The rule states that to differentiate \(x^n\), you multiply the exponent \(n\) by the variable raised to one less than the original power, giving you \(nx^{n-1}\).

For example, if you have \(x^3\), applying the power rule would give us the derivative \(3x^{3-1} = 3x^2\). This technique is fundamental when differentiating polynomials term by term, which is what you would do after expanding the expression in the original exercise. The power rule is simple yet powerful, allowing for quick differentiation without the need for more complex rules when dealing with polynomials.

Product Rule

Unlike the power rule that applies to single terms, the product rule is used when differentiating products of two functions. Rather than expanding out the product before differentiating, the product rule provides a direct method to find the derivative. This rule states that the derivative of a product \(uv\), where \(u\) and \(v\) are both functions of \(x\), is \(u'v + uv'\). This means you take the derivative of the first function \(u'\) and multiply it by the second function \(v\), then add the product of the first function \(u\) and the derivative of the second function \(v'\).

Using the product rule can be especially beneficial when the functions are complicated or would result in a cumbersome polynomial after expansion. It is also a great check for confirming the accuracy of the derivative obtained by the power rule after expansion, as seen in the exercise, illustrating the coherence of calculus concepts.