Understanding Derivatives and the Product Rule

The video tutorial explains how to determine the derivative of a function that is a product of two functions: 6 to the power of x and log base 3 of x. It introduces the product rule and the derivative formulas for exponential and logarithmic functions. The tutorial walks through setting up the product rule, applying it, and calculating the derivatives of the individual functions. Finally, it combines these derivatives to form the complete derivative function, providing a clear example of the process.

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

• Differentiate each function separately

• Use the quotient rule

• Recognize the product of two functions

• Apply the chain rule

Which rule is used to find the derivative of a product of two functions?

• Chain rule

• Power rule

• Quotient rule

• Product rule

In the product rule, if F and G are two functions, what is the formula for their derivative?

• F' x G + F x G'

• F x G' + G x F'

• F' x G' + F x G

• F x G + F' x G'

What is the derivative of log base 3 of x?

• 1 / (x ln 3)

• ln 3 / x

• ln x / 3

• x / ln 3

What is the derivative of 6 to the power of x?

• 6^x ln 6

• x 6^x

• ln 6 / 6^x

• 6^x / ln 6

When calculating the products in the derivative, what is the numerator of the first product?

• log base 3 of x

• ln 6

• 6^x

• x ln 3

What is the denominator of the first product in the derivative calculation?

• 6^x

• ln 3

• x ln 6

• x ln 3

Why might it be beneficial to leave the derivative in its current form rather than simplifying further?

• It is easier to integrate

• It is easier to differentiate again

• It avoids complex calculations

• It is more accurate

What is the final form of the derivative function?

• A simplified fraction

• A single exponential term

• A product of logarithms

• A combination of products