Understanding the Product Rule for Derivatives

The video tutorial explains how to find the derivative of a product of four functions using the product rule. It begins with an introduction to the problem and a detailed explanation of the product rule for derivatives. The instructor then applies the rule step-by-step to the given expression, demonstrating how to take derivatives of each term and simplify the resulting expression. The tutorial concludes with a summary of the process and the final simplified expression.

What is the main challenge in finding the derivative of the given expression?

• It involves a single function.

• It involves a quotient of functions.

• It involves a product of four functions.

• It involves a sum of functions.

Which rule is essential for finding the derivative of a product of functions?

• Product Rule

• Power Rule

• Chain Rule

• Quotient Rule

How is the product rule applied when there are three functions?

• Take the derivative of the first function only.

• Take the derivative of each function separately and add them.

• Take the derivative of each function while keeping the others constant, then sum the results.

• Multiply the derivatives of all functions.

What is the derivative of x squared using the power rule?

• 2x^2

• 2x

• x

• x^2

When applying the product rule to the first term, what is kept constant?

• All other terms except the one being differentiated

• e to the 4x

• Sine x

• x squared

What is the derivative of sine x?

• Sine x

• Cosine x

• Negative sine x

• Negative cosine x

What is the derivative of e to the 4x?

• e to the x

• 4x

• 4e to the 4x

• e to the 4x

What is the derivative of the natural log of x?

• ln(x)

• x ln(x)

• x

• 1/x

What is the greatest common factor that can be factored out from the expression?

• Natural log of x

• e to the 4x

• Sine x

• x squared

What is the final step in simplifying the expression?

• Adding all terms together

• Multiplying all terms

• Factoring out common terms

• Dividing by x