Understanding Derivatives and Rules

The video tutorial explains how to determine the derivative of a given function using the product rule and chain rule. It begins by identifying the function as a product of two functions and explains the need to apply the product rule. The tutorial then discusses composite functions and the application of the chain rule. Detailed steps are provided to find the derivatives, followed by algebraic simplification of the expression. The tutorial concludes with the final form of the derivative, emphasizing the importance of algebra in solving such problems.

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

• Apply the chain rule

• Identify the functions and apply the product rule

• Use the quotient rule

• Differentiate each function separately

Why is the chain rule necessary for differentiating the second function?

• Because it is a linear function

• Because it is a constant function

• Because it is a composite function

• Because it is a simple function

How is the function rewritten using rational exponents?

• As a square

• As a power of 1/2

• As a logarithm

• As a fraction

What is the purpose of applying the product rule in this context?

• To integrate the functions

• To differentiate the product of two functions

• To find the product of the derivatives

• To find the sum of the functions

What is the derivative of the inner function U in terms of X?

• 6X

• 2X

• 4X

• 8X

What is the result of differentiating U to the 1/2 with respect to U?

• 1/2 U to the 1/2

• U to the -1/2

• U to the 1/2

• 1/2 U to the -1/2

What is the next step after finding the derivatives of the functions?

• Rewrite the functions

• Apply the quotient rule

• Simplify the expression using algebra

• Integrate the functions

How is the negative exponent handled in the simplification process?

• It is multiplied by 2

• It is ignored

• It is moved to the numerator

• It is moved to the denominator

What is the common denominator used for in the final simplification?

• To multiply the fractions

• To divide the fractions

• To add the fractions

• To subtract the fractions

What is the greatest common factor of the terms in the numerator?

• 6X

• 12X

• 24X

• 18X