Understanding Derivatives and the Product Rule

The video tutorial explains how to find the derivative of a function using the product rule. It begins by identifying the need for the product rule when dealing with a product of two functions. The tutorial then sets up the derivative function and applies the chain rule to differentiate complex expressions. Finally, it simplifies and factors the derivative expression, providing both expanded and factored forms. The tutorial aims to help viewers understand the process of differentiation using these rules.

What is the first step in finding the derivative of a function that is a product of two functions?

• Apply the chain rule

• Use the quotient rule

• Differentiate each function separately

• Identify the two functions and apply the product rule

According to the product rule, what is the derivative of a product of two functions U and V?

• U * V + U' * V'

• U' * V' + U * V

• U * V' + V * U'

• U' * V + U * V'

In the given problem, what are the two functions U and V?

• U = x^2, V = e^x

• U = x^2, V = e^4.5x

• U = x^3, V = e^4.5x

• U = x^3, V = e^x

What is the derivative of x^3?

• 3x^2

• x^2

• 3x

• x^3

Which rule is applied to differentiate e^(4.5x)?

• Product rule

• Quotient rule

• Chain rule

• Power rule

What is the derivative of e^(4.5x) using the chain rule?

• 4.5x * e^(4.5x)

• e^(4.5x)

• 4.5 * e^(4.5x)

• e^(x)

What is the simplified form of the derivative function after applying the product and chain rules?

• 4.5x^2 * e^(4.5x) + 3x^3 * e^(4.5x)

• 4.5x^2 * e^(x) + 3x^3 * e^(x)

• 4.5x^3 * e^(4.5x) + 3x^2 * e^(4.5x)

• 4.5x^3 * e^(x) + 3x^2 * e^(x)

What is the greatest common factor in the derivative expression?

• e^(4.5x)

• x * e^(4.5x)

• x^2 * e^(4.5x)

• x^3 * e^(4.5x)

After factoring, what is the expression inside the parentheses?

• 4.5x^3 + 3x^2

• 4.5x + 3

• 4.5x^2 + 3x

• 4.5 + 3x

Which of the following is an acceptable form of the derivative function?

• 4.5x^2 * e^(x) + 3x^3 * e^(x)

• 4.5x^3 * e^(4.5x) + 3x^2 * e^(4.5x)

• 4.5x^2 * e^(4.5x) + 3x^3 * e^(4.5x)

• 4.5x^3 * e^(x) + 3x^2 * e^(x)