Derivatives and Product Rule Concepts

The video tutorial explains how to find the derivative of functions using the product rule. It begins with an introduction to derivatives and the product rule, followed by a detailed application of the rule to the function f(x) = x^(-5) * e^x. The tutorial demonstrates how to simplify the derivative and express it in different forms. It then moves on to finding the derivative of another function, g(x), using the same rule, and shows how to express g'(x) in various forms. The video emphasizes understanding the process and offers multiple ways to represent the derivative.

What rule is applied to find the derivative of a product of two functions?

• Product Rule

• Chain Rule

• Quotient Rule

• Power Rule

In the product rule, if u and v are functions of x, what is the formula for the derivative of u*v?

• u' * v + u * v'

• u * v'

• u' * v + v' * u

• u' * v

What is the first function (u) in the given function f(x) = x^(-5) * e^x?

• e^x

• x^5

• e^(-x)

• x^(-5)

What is the derivative of e^x with respect to x?

• e^x

• x * e^x

• 1

• x

What is the derivative of x^(-5) with respect to x?

• -5 * x^(-6)

• 5 * x^(-4)

• 5 * x^(-6)

• -5 * x^(-4)

Tags

CCSS.HSF-BF.A.1C

Which of the following is a simplified form of the derivative of f(x) = x^(-5) * e^x?

• x^(-5) * e^x - 5 * x^(-5) * e^x

• x^(-5) * e^x + 5 * x^(-6) * e^x

• x^(-5) * e^x + 5 * x^(-5) * e^x

• x^(-5) * e^x - 5 * x^(-6) * e^x

Tags

CCSS.HSF-BF.A.1C

What is the first function (u) in the given function g(x) = (3x^2 - 2) * e^x?

• e^x

• 3x^2

• x^2

• 3x^2 - 2

What is the derivative of 3x^2 - 2 with respect to x?

• 6x

• 6x - 2

• 3x^2

• 3x

Which of the following is a simplified form of the derivative of g(x) = (3x^2 - 2) * e^x?

• 3x^2 * e^x - 2 * e^x + 6x * e^x

• 3x^2 * e^x - 2 * e^x - 6x * e^x

• 3x^2 * e^x + 2 * e^x - 6x * e^x

• 3x^2 * e^x + 2 * e^x + 6x * e^x

Which of the following is a factored form of the derivative of g(x) = (3x^2 - 2) * e^x?

• e^x * (3x^2 + 6x - 2)

• e^x * (3x^2 - 6x - 2)

• e^x * (3x^2 + 6x + 2)

• e^x * (3x^2 - 6x + 2)