Understanding Derivatives and Product Rule

The video tutorial explains how to find the derivative of a given function using the product rule. It begins by introducing the function and the problem statement, followed by a detailed explanation of the product rule for differentiation. The tutorial then walks through the step-by-step calculation of derivatives, including simplification of the derivative function. Different forms of the derivative are presented, and the tutorial concludes with evaluating the derivative at X=1.

What is the initial form of the function given in the video?

• F(x) = 3x^12 * ln(x)

• F(x) = 3x^2 * ln(x)

• F(x) = 3x^1/2 * ln(x)

• F(x) = 3x^3 * ln(x)

Which rule is applied to differentiate the product of two functions?

• Chain Rule

• Power Rule

• Quotient Rule

• Product Rule

In the product rule, if U is 3x^1/2, what is V?

• 1/x

• 3x^1/2

• ln(x)

• x^2

What is the derivative of ln(x) with respect to x?

• x

• 1/x

• ln(x)

• x^2

How is the expression 3x^12 simplified after applying the product rule?

• 3x^2

• 3x^-2

• 3x^1/2

• 3x^3

Which of the following is a possible form of the derivative function?

• 3x^1/2 + ln(x)/x^2

• 3x^3 + ln(x)/x^1/2

• 3x^2 + ln(x)/x

• 3/x^2 + 3ln(x)/2x^1/2

What is the common denominator used to combine the fractions in the derivative function?

• x^2

• 2x^1/2

• 2x

• x^1/2

What is the value of ln(1) used in the substitution?

• 0

• ln(1)

• 1

• Undefined

What is the final simplified value of the derivative when x=1?

• 4

• 2

• 1

• 3

Which form of the derivative function is used for substitution?

• 3/x^2 + 3ln(x)/2x^1/2

• 6 + 3ln(x) / 2x^1/2

• 3x^2 + ln(x)/x

• 3x^1/2 + ln(x)/x^2