Understanding Derivatives from a Video Tutorial

The video tutorial explains how to find the first and second derivatives of the function F(x) = 3 ln(x). It begins by calculating the first derivative, F'(x), using the derivative of the natural logarithm, resulting in 3/x. The tutorial then evaluates F'(6) by substituting x with 6, simplifying to 2. Next, it proceeds to find the second derivative, F''(x), by applying the power rule to the first derivative, resulting in -3/x^2. Finally, it evaluates F''(6) by substituting x with 6, simplifying to -1/12. The tutorial concludes with a brief summary of the steps taken.

What is the given function F(x) in the problem?

• F(x) = 3x

• F(x) = 3 ln(x)

• F(x) = ln(3x)

• F(x) = 3/x

What is the derivative of the natural logarithm function ln(x)?

• 1/x

• x

• ln(x)

• x^2

How is the first derivative of F(x) = 3 ln(x) expressed?

• 3x^2

• ln(3x)

• 3/x

• 3x

What is the value of the first derivative at x = 6?

• 3

• 1

• 6

• 2

How can 3/x be rewritten to apply the power rule?

• 3x^1

• 3x^2

• 3x^-1

• 3x^0

What is the second derivative of F(x) = 3 ln(x)?

• -3/x^2

• 3x

• 3/x^2

• -3x

Which rule is used to find the second derivative from the first derivative?

• Chain Rule

• Product Rule

• Quotient Rule

• Power Rule

What is the value of the second derivative at x = 6?

• -1/6

• -1/12

• -1/18

• -1/36

What is the common factor used to simplify -3/36?

• 3

• 6

• 12

• 2