Calculus I: The Product Rule (Level 1 of 3)

The video tutorial explores the concept of derivatives, focusing on the product rule. It begins by questioning if the derivative of a product is the product of derivatives, which is disproven. The product rule is introduced, stating that the derivative of a product of two functions is the first function times the derivative of the second plus the second function times the derivative of the first. Examples are provided, including products of more than two functions, and the application of the product rule to quotients. The tutorial concludes with complex examples, emphasizing the importance of consistency in function assignment.

What is the derivative of a sum or difference of functions equal to?

• The square of the derivatives of the functions

• The sum or difference of the derivatives of the functions

• The product of the derivatives of the functions

• The quotient of the derivatives of the functions

Why is the derivative of a product not simply the product of the derivatives?

• Because it is equal to zero

• Because it is undefined

• Because it requires the use of the product rule

• Because it involves a sum of derivatives

What does the product rule state for the derivative of two functions?

• The product of the derivatives of both functions

• The derivative of the first function times the second function

• The first function times the derivative of the second function plus the second function times the derivative of the first function

• The sum of the derivatives of both functions

In the example of finding the derivative of R(x) = x^3 * e^x, what is the first step?

• Differentiate each term separately

• Assign x^3 as F(x) and e^x as G(x)

• Use the chain rule

• Apply the quotient rule

How can you rewrite a quotient to apply the product rule?

• By using the chain rule

• By expressing the denominator as a negative exponent

• By adding a constant

• By multiplying by zero

What is the derivative of R(x) = e^x * (x^2 - 2x) using the product rule?

• e^x * (x^2 - 2x) - 2

• e^x * (x^2 - 2)

• e^x * (x^2 + 2x)

• e^x * (x^2 - 2x) + 2

When applying the product rule to three functions, what is important to remember?

• To differentiate all functions simultaneously

• To ignore the order of functions

• To only differentiate the first function

• To alternate the derivative among the functions

What is a benefit of using the extended product rule for more than two functions?

• It eliminates the need for any derivatives

• It provides an exact solution without simplification

• It allows for easier differentiation of non-polynomial expressions

• It simplifies the expression to a single term

What is the final derivative of P(x) = (x^3 - x)(x^2 + 2)(x^2 + x - 1) after simplification?

• 7x^6 + 6x^5 + 4x^3 - 9x^2 - 4x + 2

• x^6 + x^5 + x^3 - x^2 - x + 1

• x^6 - x^5 + x^3 - x^2 + x - 1

• 7x^6 - 6x^5 + 4x^3 + 9x^2 + 4x - 2

What should you do if you want to practice distribution and collecting like terms?

• Only differentiate the first term

• Use the chain rule

• Simplify the expression after applying the product rule

• Ignore the product rule