Calculating Derivatives and Tangent Lines

The video tutorial explains how to find the derivative of a composite function F(x) = g(x)^3 at x=4. It involves using the chain rule to differentiate, substituting given values, and calculating the slope of the tangent line to find F prime of 4, which is -54.

What is the expression for F(x) given in the problem?

• F(x) = g(x) + 3

• F(x) = g(x)^2

• F(x) = g(x)^3

• F(x) = 3g(x)

What is the main goal of the problem?

• To find the integral of F(x)

• To find the value of g(x) at x = 4

• To find the slope of the tangent line

• To find the value of F'(4)

Which rule is primarily used to find the derivative of g(x)^3?

• Sum Rule

• Product Rule

• Quotient Rule

• Chain Rule

What is the derivative of x^3 with respect to x?

• 3x

• x^3

• x^2

• 3x^2

Which mathematical concept is used to find the derivative of a composite function?

• Chain Rule

• Quotient Rule

• Product Rule

• Power Rule

What is the value of g(4) as given in the problem?

• 3

• 2

• 5

• 4

What does g'(4) represent in the context of the problem?

• The second derivative of g(x) at x = 4

• The integral of g(x) at x = 4

• The slope of the tangent line at x = 4

• The value of g(x) at x = 4

How is the slope of the tangent line calculated?

• By adding x and y

• By finding the change in x over change in y

• By multiplying x and y

• By finding the change in y over change in x

What is the slope of the tangent line at x = 4?

• 2

• 4

• -4

• -2

What is the final value of F'(4)?

• 27

• -27

• 54

• -54